Daily Papers

In this work, we develop language models for the Sanskrit language, namely Bidirectional Encoder Representations from Transformers (BERT) and its variants: A Lite BERT (ALBERT), and Robustly Optimized BERT (RoBERTa) using Devanagari Sanskrit text corpus. Then we extracted the features for the given text from these models. We applied the dimensional reduction and clustering techniques on the features to generate an extractive summary for a given Sanskrit document. Along with the extractive text summarization techniques, we have also created and released a Sanskrit Devanagari text corpus publicly.

Reduction in effective spacetime dimensionality can occur in field-theory models more general than the widely studied dimensional reductions based on technically consistent truncations. Situations where wavefunction factors depend nontrivially on coordinates transverse to the effective lower dimension can give rise to unusual patterns of gauge symmetry breaking. Leading-order gauge modes can be left massless, but naturally occurring Stueckelberg modes can couple importantly at quartic order and higher, thus generating a "covert" pattern of gauge symmetry breaking. Such a situation is illustrated in a five-dimensional model of scalar electrodynamics in which one spatial dimension is taken to be an interval with Dirichlet/Robin boundary conditions on opposing ends. This simple model illuminates a mechanism which also has been found in gravitational braneworld scenarios.

Conventional urban indicators derived from censuses, surveys, and administrative records are often costly, spatially inconsistent, and slow to update. Recent geospatial foundation models enable Earth embeddings, compact satellite image representations transferable across downstream tasks, but their utility for neighborhood-scale urban monitoring remains unclear. Here, we benchmark three Earth embedding families, AlphaEarth, Prithvi, and Clay, for urban signal prediction across six U.S. metropolitan areas from 2020 to 2023. Using a unified supervised-learning framework, we predict 14 neighborhood-level indicators spanning crime, income, health, and travel behavior, and evaluate performance under four settings: global, city-wise, year-wise, and city-year. Results show that Earth embeddings capture substantial urban variation, with the highest predictive skill for outcomes more directly tied to built-environment structure, including chronic health burdens and dominant commuting modes. By contrast, indicators shaped more strongly by fine-scale behavior and local policy, such as cycling, remain difficult to infer. Predictive performance varies markedly across cities but remains comparatively stable across years, indicating strong spatial heterogeneity alongside temporal robustness. Exploratory analysis suggests that cross-city variation in predictive performance is associated with urban form in task-specific ways. Controlled dimensionality experiments show that representation efficiency is critical: compact 64-dimensional AlphaEarth embeddings remain more informative than 64-dimensional reductions of Prithvi and Clay. This study establishes a benchmark for evaluating Earth embeddings in urban remote sensing and demonstrates their potential as scalable, low-cost features for SDG-aligned neighborhood-scale urban monitoring.

In complex oxide materials, changes in electronic properties are often associated with changes in crystal structure, raising the question of the relative roles of the electronic and lattice effects in driving the metal-insulator transition. This paper presents a combined theoretical and experimental analysis of the dependence of the metal-insulator transition of NdNiO_3 on crystal structure, specifically comparing properties of bulk materials to one and two layer samples of NdNiO_3 grown between multiple electronically inert NdAlO_3 counterlayers in a superlattice. The comparison amplifies and validates a theoretical approach developed in previous papers and disentangles the electronic and lattice contributions, through an independent variation of each. In bulk NdNiO_3 the correlations are not strong enough to drive a metal-insulator transition by themselves: a lattice distortion is required. Ultra-thin films exhibit two additional electronic effects and one lattice-related effect. The electronic effects are quantum confinement, leading to dimensional reduction of the electronic Hamiltonian, and an increase in electronic bandwidth due to counterlayer induced bond angle changes. We find that the confinement effect is much more important. The lattice effect is an increase in stiffness due to the cost of propagation of the lattice disproportionation into the confining material.

Real-time detection and mitigation of technical anomalies are critical for large-scale cloud-native services, where even minutes of downtime can result in massive financial losses and diminished user trust. While customer incidents serve as a vital signal for discovering risks missed by monitoring, extracting actionable intelligence from this data remains challenging due to extreme noise, high throughput, and semantic complexity of diverse business lines. In this paper, we present TingIS, an end-to-end system designed for enterprise-grade incident discovery. At the core of TingIS is a multi-stage event linking engine that synergizes efficient indexing techniques with Large Language Models (LLMs) to make informed decisions on event merging, enabling the stable extraction of actionable incidents from just a handful of diverse user descriptions. This engine is complemented by a cascaded routing mechanism for precise business attribution and a multi-dimensional noise reduction pipeline that integrates domain knowledge, statistical patterns, and behavioral filtering. Deployed in a production environment handling a peak throughput of over 2,000 messages per minute and 300,000 messages per day, TingIS achieves a P90 alert latency of 3.5 minutes and a 95\% discovery rate for high-priority incidents. Benchmarks constructed from real-world data demonstrate that TingIS significantly outperforms baseline methods in routing accuracy, clustering quality, and Signal-to-Noise Ratio.

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

This paper presents a low-dimensional observer design for stable, single-input single-output, continuous-time linear time-invariant (LTI) systems. Leveraging the model reduction by moment matching technique, we approximate the system with a reduced-order model. Based on this reduced-order model, we design a low-dimensional observer that estimates the states of the original system. We show that this observer establishes exact asymptotic state reconstruction for a given class of inputs tied to the observer's dimension. Furthermore, we establish an exponential input-to-state stability property for generic inputs, ensuring a bounded estimation error. Numerical simulations confirm the effectiveness of the approach for a benchmark model reduction problem.

High-dimensional vector embeddings are widely used in retrieval systems, yet dimensionality reduction (DR) is seldom applied due to its tendency to distort nearest-neighbor (NN) structure critical for search. Existing DR techniques such as PCA and UMAP optimize global or manifold-preserving criteria, rather than retrieval-specific objectives. We present MPAD: Maximum Pairwise Absolute Difference, an unsupervised DR method that explicitly preserves approximate NN relations by maximizing the margin between k-NNs and non-k-NNs under a soft orthogonality constraint. This design enables MPAD to retain ANN-relevant geometry without supervision or changes to the original embedding model. Experiments across multiple domains show that MPAD consistently outperforms standard DR methods in preserving neighborhood structure, enabling more accurate search in reduced dimensions.

Semantic representations of text, i.e. representations of natural language which capture meaning by geometry, are essential for areas such as information retrieval and document grouping. High-dimensional trained dense vectors have received much attention in recent years as such representations. We investigate the structure of semantic spaces that arise from embeddings made with Sentence-BERT and find that the representations suffer from a well-known problem in high dimensions called hubness. Hubness results in asymmetric neighborhood relations, such that some texts (the hubs) are neighbours of many other texts while most texts (so-called anti-hubs), are neighbours of few or no other texts. We quantify the semantic quality of the embeddings using hubness scores and error rate of a neighbourhood based classifier. We find that when hubness is high, we can reduce error rate and hubness using hubness reduction methods. We identify a combination of two methods as resulting in the best reduction. For example, on one of the tested pretrained models, this combined method can reduce hubness by about 75% and error rate by about 9%. Thus, we argue that mitigating hubness in the embedding space provides better semantic representations of text.

Video generative models can be regarded as world simulators due to their ability to capture dynamic, continuous changes inherent in real-world environments. These models integrate high-dimensional information across visual, temporal, spatial, and causal dimensions, enabling predictions of subjects in various status. A natural and valuable research direction is to explore whether a fully trained video generative model in high-dimensional space can effectively support lower-dimensional tasks such as controllable image generation. In this work, we propose a paradigm for video-to-image knowledge compression and task adaptation, termed Dimension-Reduction Attack (DRA-Ctrl), which utilizes the strengths of video models, including long-range context modeling and flatten full-attention, to perform various generation tasks. Specially, to address the challenging gap between continuous video frames and discrete image generation, we introduce a mixup-based transition strategy that ensures smooth adaptation. Moreover, we redesign the attention structure with a tailored masking mechanism to better align text prompts with image-level control. Experiments across diverse image generation tasks, such as subject-driven and spatially conditioned generation, show that repurposed video models outperform those trained directly on images. These results highlight the untapped potential of large-scale video generators for broader visual applications. DRA-Ctrl provides new insights into reusing resource-intensive video models and lays foundation for future unified generative models across visual modalities. The project page is https://dra-ctrl-2025.github.io/DRA-Ctrl/.

Four-dimensional cone-beam computed tomography (4D CBCT) provides respiration-resolved images and can be used for image-guided radiation therapy. However, the ability to reveal respiratory motion comes at the cost of image artifacts. As raw projection data are sorted into multiple respiratory phases, the cone-beam projections become much sparser and the reconstructed 4D CBCT images will be covered by severe streak artifacts. Although several deep learning-based methods have been proposed to address this issue, most algorithms employ 2D network models as backbones, neglecting the intrinsic structural priors within 4D CBCT images. In this paper, we first explore the origin and appearance of streak artifacts in 4D CBCT images. We find that streak artifacts exhibit a unique rotational motion along with the patient's respiration, distinguishable from diaphragm-driven respiratory motion in the spatiotemporal domain. Therefore, we propose a novel 4D neural network model, RSTAR4D-Net, designed to address Rotational STreak Artifact Reduction by integrating the spatial and temporal information within 4D CBCT images. Specifically, we overcome the computational and training difficulties of a 4D neural network. The specially designed model adopts an efficient implementation of 4D convolutions to reduce computational costs and thus can process the whole 4D image in one pass. Additionally, a Tetris training strategy pertinent to the separable 4D convolutions is proposed to effectively train the model using limited 4D training samples. Extensive experiments substantiate the effectiveness of our proposed method, and the RSTAR4D-Net shows superior performance compared to other methods. The source code and dynamic demos are available at https://github.com/ivy9092111111/RSTAR.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.

Dimensionality reduction methods are unsupervised approaches which learn low-dimensional spaces where some properties of the initial space, typically the notion of "neighborhood", are preserved. Such methods usually require propagation on large k-NN graphs or complicated optimization solvers. On the other hand, self-supervised learning approaches, typically used to learn representations from scratch, rely on simple and more scalable frameworks for learning. In this paper, we propose TLDR, a dimensionality reduction method for generic input spaces that is porting the recent self-supervised learning framework of Zbontar et al. (2021) to the specific task of dimensionality reduction, over arbitrary representations. We propose to use nearest neighbors to build pairs from a training set and a redundancy reduction loss to learn an encoder that produces representations invariant across such pairs. TLDR is a method that is simple, easy to train, and of broad applicability; it consists of an offline nearest neighbor computation step that can be highly approximated, and a straightforward learning process. Aiming for scalability, we focus on improving linear dimensionality reduction, and show consistent gains on image and document retrieval tasks, e.g. gaining +4% mAP over PCA on ROxford for GeM- AP, improving the performance of DINO on ImageNet or retaining it with a 10x compression.

Across many scientific fields, measurements often represent the number of times an event occurs. For example, a document can be represented by word occurrence counts, neural activity by spike counts per time window, or online communication by daily email counts. These measurements yield high-dimensional count data that often approximate a Poisson distribution, frequently with low rates that produce substantial sparsity and complicate downstream analysis. A useful approach is to embed the data into a low-dimensional space that preserves meaningful structure, commonly termed dimensionality reduction. Yet existing dimensionality reduction methods, including both linear (e.g., PCA) and nonlinear approaches (e.g., t-SNE), often assume continuous Euclidean geometry, thereby misaligning with the discrete, sparse nature of low-rate count data. Here, we propose p-SNE (Poisson Stochastic Neighbor Embedding), a nonlinear neighbor embedding method designed around the Poisson structure of count data, using KL divergence between Poisson distributions to measure pairwise dissimilarity and Hellinger distance to optimize the embedding. We test p-SNE on synthetic Poisson data and demonstrate its ability to recover meaningful structure in real-world count datasets, including weekday patterns in email communication, research area clusters in OpenReview papers, and temporal drift and stimulus gradients in neural spike recordings.

Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents a nonlinear model reduction method that employs lifting variable transformations to derive structure-preserving quadratic reduced-order models for conservative PDEs with general nonlinearities. We present an energy-quadratization strategy that defines the auxiliary variable in terms of the nonlinear term in the energy expression to derive an equivalent quadratic lifted system with quadratic system energy. The proposed strategy combined with proper orthogonal decomposition model reduction yields quadratic reduced-order models that conserve the quadratized lifted energy exactly in high dimensions. We demonstrate the proposed model reduction approach on four nonlinear conservative PDEs: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, the two-dimensional Klein-Gordon equation with parametric dependence, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed lifting approach is competitive with the state-of-the-art structure-preserving hyper-reduction method in terms of both accuracy and computational efficiency in the online stage while providing significant computational gains in the offline stage.

Scaling multi-dimensional transformers to long sequences is indispensable across various domains. However, the challenges of large memory requirements and slow speeds of such sequences necessitate sequence parallelism. All existing approaches fall under the category of embedded sequence parallelism, which are limited to shard along a single sequence dimension, thereby introducing significant communication overhead. However, the nature of multi-dimensional transformers involves independent calculations across multiple sequence dimensions. To this end, we propose Dynamic Sequence Parallelism (DSP) as a novel abstraction of sequence parallelism. DSP dynamically switches the parallel dimension among all sequences according to the computation stage with efficient resharding strategy. DSP offers significant reductions in communication costs, adaptability across modules, and ease of implementation with minimal constraints. Experimental evaluations demonstrate DSP's superiority over state-of-the-art embedded sequence parallelism methods by remarkable throughput improvements ranging from 32.2% to 10x, with less than 25% communication volume.

High-dimensional datasets require effective feature selection to improve predictive performance, interpretability, and robustness. We propose and evaluate feature selection methods for tabular datasets based on Kolmogorov-Arnold networks (KANs), which parameterize feature transformations through splines, enabling direct access to interpretable importance measures. We introduce four KAN-based selectors (KAN-L1, KAN-L2, KAN-SI, KAN-KO) and compare them against classical baselines (LASSO, Random Forest, Mutual Information, SVM-RFE) across multiple classification and regression tabular dataset benchmarks. Average (over three retention levels: 20\%, 40\%, and 60\%) F1 scores and R^2 score results reveal that KAN-based selectors, particularly KAN-L2, KAN-L1, KAN-SI, and KAN-KO, are competitive with and sometimes superior to classical baselines in structured and synthetic datasets. However, KAN-L1 is often too aggressive in regression, removing useful features, while KAN-L2 underperforms in classification, where simple coefficient shrinkage misses complex feature interactions. KAN-L2 and KAN-SI provide robust performance on noisy regression datasets and heterogeneous datasets, aligning closely with ensemble predictors. In classification tasks, KAN selectors such as KAN-L1, KAN-KO, and KAN-SI sometimes surpass the other selectors by eliminating redundancy, particularly in high-dimensional multi-class data. Overall, our findings demonstrate that KAN-based feature selection provides a powerful and interpretable alternative to traditional methods, capable of uncovering nonlinear and multivariate feature relevance beyond sparsity or impurity-based measures.

Self-supervised learning (SSL) is rapidly closing the gap with supervised methods on large computer vision benchmarks. A successful approach to SSL is to learn embeddings which are invariant to distortions of the input sample. However, a recurring issue with this approach is the existence of trivial constant solutions. Most current methods avoid such solutions by careful implementation details. We propose an objective function that naturally avoids collapse by measuring the cross-correlation matrix between the outputs of two identical networks fed with distorted versions of a sample, and making it as close to the identity matrix as possible. This causes the embedding vectors of distorted versions of a sample to be similar, while minimizing the redundancy between the components of these vectors. The method is called Barlow Twins, owing to neuroscientist H. Barlow's redundancy-reduction principle applied to a pair of identical networks. Barlow Twins does not require large batches nor asymmetry between the network twins such as a predictor network, gradient stopping, or a moving average on the weight updates. Intriguingly it benefits from very high-dimensional output vectors. Barlow Twins outperforms previous methods on ImageNet for semi-supervised classification in the low-data regime, and is on par with current state of the art for ImageNet classification with a linear classifier head, and for transfer tasks of classification and object detection.

To learn intrinsic low-dimensional structures from high-dimensional data that most discriminate between classes, we propose the principle of Maximal Coding Rate Reduction (MCR^2), an information-theoretic measure that maximizes the coding rate difference between the whole dataset and the sum of each individual class. We clarify its relationships with most existing frameworks such as cross-entropy, information bottleneck, information gain, contractive and contrastive learning, and provide theoretical guarantees for learning diverse and discriminative features. The coding rate can be accurately computed from finite samples of degenerate subspace-like distributions and can learn intrinsic representations in supervised, self-supervised, and unsupervised settings in a unified manner. Empirically, the representations learned using this principle alone are significantly more robust to label corruptions in classification than those using cross-entropy, and can lead to state-of-the-art results in clustering mixed data from self-learned invariant features.

In this paper, we contend that a natural objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a low-dimensional Gaussian mixture supported on incoherent subspaces. The goodness of such a representation can be evaluated by a principled measure, called sparse rate reduction, that simultaneously maximizes the intrinsic information gain and extrinsic sparsity of the learned representation. From this perspective, popular deep network architectures, including transformers, can be viewed as realizing iterative schemes to optimize this measure. Particularly, we derive a transformer block from alternating optimization on parts of this objective: the multi-head self-attention operator compresses the representation by implementing an approximate gradient descent step on the coding rate of the features, and the subsequent multi-layer perceptron sparsifies the features. This leads to a family of white-box transformer-like deep network architectures, named CRATE, which are mathematically fully interpretable. We show, by way of a novel connection between denoising and compression, that the inverse to the aforementioned compressive encoding can be realized by the same class of CRATE architectures. Thus, the so-derived white-box architectures are universal to both encoders and decoders. Experiments show that these networks, despite their simplicity, indeed learn to compress and sparsify representations of large-scale real-world image and text datasets, and achieve performance very close to highly engineered transformer-based models: ViT, MAE, DINO, BERT, and GPT2. We believe the proposed computational framework demonstrates great potential in bridging the gap between theory and practice of deep learning, from a unified perspective of data compression. Code is available at: https://ma-lab-berkeley.github.io/CRATE .

In this paper, we contend that the objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a mixture of low-dimensional Gaussian distributions supported on incoherent subspaces. The quality of the final representation can be measured by a unified objective function called sparse rate reduction. From this perspective, popular deep networks such as transformers can be naturally viewed as realizing iterative schemes to optimize this objective incrementally. Particularly, we show that the standard transformer block can be derived from alternating optimization on complementary parts of this objective: the multi-head self-attention operator can be viewed as a gradient descent step to compress the token sets by minimizing their lossy coding rate, and the subsequent multi-layer perceptron can be viewed as attempting to sparsify the representation of the tokens. This leads to a family of white-box transformer-like deep network architectures which are mathematically fully interpretable. Despite their simplicity, experiments show that these networks indeed learn to optimize the designed objective: they compress and sparsify representations of large-scale real-world vision datasets such as ImageNet, and achieve performance very close to thoroughly engineered transformers such as ViT. Code is at https://github.com/Ma-Lab-Berkeley/CRATE.

Retrieval-Augmented Generation enhances language models by retrieving relevant information from external knowledge bases, relying on high-dimensional vector embeddings typically stored in float32 precision. However, storing these embeddings at scale presents significant memory challenges. To address this issue, we systematically investigate on MTEB benchmark two complementary optimization strategies: quantization, evaluating standard formats (float16, int8, binary) and low-bit floating-point types (float8), and dimensionality reduction, assessing methods like PCA, Kernel PCA, UMAP, Random Projections and Autoencoders. Our results show that float8 quantization achieves a 4x storage reduction with minimal performance degradation (<0.3%), significantly outperforming int8 quantization at the same compression level, being simpler to implement. PCA emerges as the most effective dimensionality reduction technique. Crucially, combining moderate PCA (e.g., retaining 50% dimensions) with float8 quantization offers an excellent trade-off, achieving 8x total compression with less performance impact than using int8 alone (which provides only 4x compression). To facilitate practical application, we propose a methodology based on visualizing the performance-storage trade-off space to identify the optimal configuration that maximizes performance within their specific memory constraints.

Nonlinear sufficient dimension reductionlibing_generalSDR, which constructs nonlinear low-dimensional representations to summarize essential features of high-dimensional data, is an important branch of representation learning. However, most existing methods are not applicable when the response variables are complex non-Euclidean random objects, which are frequently encountered in many recent statistical applications. In this paper, we introduce a new statistical dependence measure termed Fr\'echet Cumulative Covariance (FCCov) and develop a novel nonlinear SDR framework based on FCCov. Our approach is not only applicable to complex non-Euclidean data, but also exhibits robustness against outliers. We further incorporate Feedforward Neural Networks (FNNs) and Convolutional Neural Networks (CNNs) to estimate nonlinear sufficient directions in the sample level. Theoretically, we prove that our method with squared Frobenius norm regularization achieves unbiasedness at the sigma-field level. Furthermore, we establish non-asymptotic convergence rates for our estimators based on FNNs and ResNet-type CNNs, which match the minimax rate of nonparametric regression up to logarithmic factors. Intensive simulation studies verify the performance of our methods in both Euclidean and non-Euclidean settings. We apply our method to facial expression recognition datasets and the results underscore more realistic and broader applicability of our proposal.

Simulating physical systems governed by Lagrangian dynamics often entails solving partial differential equations (PDEs) over high-resolution spatial domains, leading to significant computational expense. Reduced-order modeling (ROM) mitigates this cost by evolving low-dimensional latent representations of the underlying system. While neural ROMs enable querying solutions from latent states at arbitrary spatial points, their latent states typically represent the global domain and struggle to capture localized, highly dynamic behaviors such as fluids. We propose a sampling-based reduction framework that evolves Lagrangian systems directly in physical space over the particles themselves, reducing the number of active degrees of freedom via data-driven neural PDE operators. To enable querying at arbitrary spatial locations, we introduce a learnable kernel parameterization that uses local spatial information from time-evolved sample particles to infer the underlying solution manifold. Empirically, our approach achieves a 6.6x to 32x reduction in input dimensionality while maintaining high-fidelity evaluations across diverse Lagrangian regimes, including fluid flows, granular media, and elastoplastic dynamics. We refer to this framework as GIOROM (Geometry-Informed Reduced-Order Modeling). All code and data are available at: https://github.com/HrishikeshVish/GIOROM

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

Anti-Money Laundering (AML) is a crucial task in ensuring the integrity of financial systems. One keychallenge in AML is identifying high-risk groups based on their behavior. Unsupervised learning, particularly clustering, is a promising solution for this task. However, the use of hundreds of features todescribe behavior results in a highdimensional dataset that negatively impacts clustering performance.In this paper, we investigate the effectiveness of combining clustering method agglomerative hierarchicalclustering with four dimensionality reduction techniques -Independent Component Analysis (ICA), andKernel Principal Component Analysis (KPCA), Singular Value Decomposition (SVD), Locality Preserving Projections (LPP)- to overcome the issue of high-dimensionality in AML data and improve clusteringresults. This study aims to provide insights into the most effective way of reducing the dimensionality ofAML data and enhance the accuracy of clustering-based AML systems. The experimental results demonstrate that KPCA outperforms other dimension reduction techniques when combined with agglomerativehierarchical clustering. This superiority is observed in the majority of situations, as confirmed by threedistinct validation indices.

Embedding nonlinear dynamical systems into artificial neural networks is a powerful new formalism for machine learning. By parameterizing ordinary differential equations (ODEs) as neural network layers, these Neural ODEs are memory-efficient to train, process time-series naturally and incorporate knowledge of physical systems into deep learning models. However, the practical applications of Neural ODEs are limited due to long inference times, because the outputs of the embedded ODE layers are computed numerically with differential equation solvers that can be computationally demanding. Here we show that mathematical model order reduction methods can be used for compressing and accelerating Neural ODEs by accurately simulating the continuous nonlinear dynamics in low-dimensional subspaces. We implement our novel compression method by developing Neural ODEs that integrate the necessary subspace-projection and interpolation operations as layers of the neural network. We validate our approach by comparing it to neuron pruning and SVD-based weight truncation methods from the literature in image and time-series classification tasks. The methods are evaluated by acceleration versus accuracy when adjusting the level of compression. On this spectrum, we achieve a favourable balance over existing methods by using model order reduction when compressing a convolutional Neural ODE. In compressing a recurrent Neural ODE, SVD-based weight truncation yields good performance. Based on our results, our integration of model order reduction with Neural ODEs can facilitate efficient, dynamical system-driven deep learning in resource-constrained applications.

Large Language Models (LLMs) excel in various natural language processing tasks but remain vulnerable to generating harmful content or being exploited for malicious purposes. Although safety alignment datasets have been introduced to mitigate such risks through supervised fine-tuning (SFT), these datasets often lack comprehensive risk coverage. Most existing datasets focus primarily on lexical diversity while neglecting other critical dimensions. To address this limitation, we propose a novel analysis framework to systematically measure the risk coverage of alignment datasets across three essential dimensions: Lexical Diversity, Malicious Intent, and Jailbreak Tactics. We further introduce TRIDENT, an automated pipeline that leverages persona-based, zero-shot LLM generation to produce diverse and comprehensive instructions spanning these dimensions. Each harmful instruction is paired with an ethically aligned response, resulting in two datasets: TRIDENT-Core, comprising 26,311 examples, and TRIDENT-Edge, with 18,773 examples. Fine-tuning Llama 3.1-8B on TRIDENT-Edge demonstrates substantial improvements, achieving an average 14.29% reduction in Harm Score, and a 20% decrease in Attack Success Rate compared to the best-performing baseline model fine-tuned on the WildBreak dataset.

Low-rank adaptation (LoRA) has become the default approach to fine-tune large language models (LLMs) due to its significant reduction in trainable parameters. However, trainable parameter demand for LoRA increases with increasing model embedding dimensions, leading to high compute costs. Additionally, its backward updates require storing high-dimensional intermediate activations and optimizer states, demanding high peak GPU memory. In this paper, we introduce large model fine-tuning via spectrally decomposed low-dimensional adaptation (LaMDA), a novel approach to fine-tuning large language models, which leverages low-dimensional adaptation to achieve significant reductions in trainable parameters and peak GPU memory footprint. LaMDA freezes a first projection matrix (PMA) in the adaptation path while introducing a low-dimensional trainable square matrix, resulting in substantial reductions in trainable parameters and peak GPU memory usage. LaMDA gradually freezes a second projection matrix (PMB) during the early fine-tuning stages, reducing the compute cost associated with weight updates to enhance parameter efficiency further. We also present an enhancement, LaMDA++, incorporating a ``lite-weight" adaptive rank allocation for the LoRA path via normalized spectrum analysis of pre-trained model weights. We evaluate LaMDA/LaMDA++ across various tasks, including natural language understanding with the GLUE benchmark, text summarization, natural language generation, and complex reasoning on different LLMs. Results show that LaMDA matches or surpasses the performance of existing alternatives while requiring up to 17.7x fewer parameter updates and up to 1.32x lower peak GPU memory usage during fine-tuning. Code will be publicly available.

Reinforcement learning often needs to deal with the exponential growth of states and actions when exploring optimal control in high-dimensional spaces (often known as the curse of dimensionality). In this work, we address this issue by learning the inherent structure of action-wise similar MDP to appropriately balance the performance degradation versus sample/computational complexity. In particular, we partition the action spaces into multiple groups based on the similarity in transition distribution and reward function, and build a linear decomposition model to capture the difference between the intra-group transition kernel and the intra-group rewards. Both our theoretical analysis and experiments reveal a surprising and counter-intuitive result: while a more refined grouping strategy can reduce the approximation error caused by treating actions in the same group as identical, it also leads to increased estimation error when the size of samples or the computation resources is limited. This finding highlights the grouping strategy as a new degree of freedom that can be optimized to minimize the overall performance loss. To address this issue, we formulate a general optimization problem for determining the optimal grouping strategy, which strikes a balance between performance loss and sample/computational complexity. We further propose a computationally efficient method for selecting a nearly-optimal grouping strategy, which maintains its computational complexity independent of the size of the action space.

Single-atom catalysts (SACs) have emerged as frontiers for catalyzing chemical reactions, yet the diverse combinations of active elements and support materials, the nature of coordination environments, elude traditional methodologies in searching optimal SAC systems with superior catalytic performance. Herein, by integrating multi-branch Convolutional Neural Network (CNN) analysis models to hybrid descriptor based activity volcano plot, 2D SAC system composed of diverse metallic single atoms anchored on six type of 2D supports, including graphitic carbon nitride, nitrogen-doped graphene, graphene with dual-vacancy, black phosphorous, boron nitride, and C2N, are screened for efficient CO2RR. Starting from establishing a correlation map between the adsorption energies of intermediates and diverse electronic and elementary descriptors, sole singular descriptor lost magic to predict catalytic activity. Deep learning method utilizing multi-branch CNN model therefore was employed, using 2D electronic density of states as input to predict adsorption energies. Hybrid-descriptor enveloping both C- and O-types of CO2RR intermediates was introduced to construct volcano plots and limiting potential periodic table, aiming for intuitive screening of catalyst candidates for efficient CO2 reduction to CH4. The eDOS occlusion experiments were performed to unravel individual orbital contribution to adsorption energy. To explore the electronic scale principle governing practical engineering catalytic CO2RR activity, orbitalwise eDOS shifting experiments based on CNN model were employed. The study involves examining the adsorption energy and, consequently, catalytic activities while varying supported single atoms. This work offers a tangible framework to inform both theoretical screening and experimental synthesis, thereby paving the way for systematically designing efficient SACs.

Understanding high-dimensional data requires projecting it into lower-dimensional spaces, but any single projection inevitably loses information or introduces distortions. Tours address this limitation through animation of 2D projection sequences, yet existing tools present tradeoffs in the freedom and steerability of projection traversal, providing little to no ability to move between expert-guided paths and unrestrained exploration. We present dtour, a tour interface that combines static projection previews, reversible scrubbing along continuous geodesic projection paths, manual projection manipulation, and a wandering grand tour, all within a single progressive exploration interface. dtour scales to millions of points via GPU-accelerated rendering, runs in any modern browser, and integrates with both Python and JavaScript ecosystems. We demonstrate dtour on text, image, and single-cell data for two usage scenarios: gradually revealing structure in high-dimensional data and validating non-linear dimensionality reduction outputs.

We develop data-driven methods incorporating geometric and topological information to learn parsimonious representations of nonlinear dynamics from observations. The approaches learn nonlinear state-space models of the dynamics for general manifold latent spaces using training strategies related to Variational Autoencoders (VAEs). Our methods are referred to as Geometric Dynamic (GD) Variational Autoencoders (GD-VAEs). We learn encoders and decoders for the system states and evolution based on deep neural network architectures that include general Multilayer Perceptrons (MLPs), Convolutional Neural Networks (CNNs), and other architectures. Motivated by problems arising in parameterized PDEs and physics, we investigate the performance of our methods on tasks for learning reduced dimensional representations of the nonlinear Burgers Equations, Constrained Mechanical Systems, and spatial fields of Reaction-Diffusion Systems. GD-VAEs provide methods that can be used to obtain representations in manifold latent spaces for diverse learning tasks involving dynamics.

Distribution matching is central to many vision and graphics tasks, where the widely used Wasserstein distance is too costly to compute for high dimensional distributions. The Sliced Wasserstein Distance (SWD) offers a scalable alternative, yet its Monte Carlo estimator suffers from high variance, resulting in noisy gradients and slow convergence. We introduce Reservoir SWD (ReSWD), which integrates Weighted Reservoir Sampling into SWD to adaptively retain informative projection directions in optimization steps, resulting in stable gradients while remaining unbiased. Experiments on synthetic benchmarks and real-world tasks such as color correction and diffusion guidance show that ReSWD consistently outperforms standard SWD and other variance reduction baselines. Project page: https://reservoirswd.github.io/

Prompt-based text embedding models, which generate task-specific embeddings upon receiving tailored prompts, have recently demonstrated remarkable performance. However, their resulting embeddings often have thousands of dimensions, leading to high storage costs and increased computational costs of embedding-based operations. In this paper, we investigate how post-hoc dimensionality reduction applied to the embeddings affects the performance of various tasks that leverage these embeddings, specifically classification, clustering, retrieval, and semantic textual similarity (STS) tasks. Our experiments show that even a naive dimensionality reduction, which keeps only the first 25% of the dimensions of the embeddings, results in a very slight performance degradation, indicating that these embeddings are highly redundant. Notably, for classification and clustering, even when embeddings are reduced to less than 0.5% of the original dimensionality the performance degradation is very small. To quantitatively analyze this redundancy, we perform an analysis based on the intrinsic dimensionality and isotropy of the embeddings. Our analysis reveals that embeddings for classification and clustering, which are considered to have very high dimensional redundancy, exhibit lower intrinsic dimensionality and less isotropy compared with those for retrieval and STS.

Bank credit risk is a significant challenge in modern financial transactions, and the ability to identify qualified credit card holders among a large number of applicants is crucial for the profitability of a bank'sbank's credit card business. In the past, screening applicants'applicants' conditions often required a significant amount of manual labor, which was time-consuming and labor-intensive. Although the accuracy and reliability of previously used ML models have been continuously improving, the pursuit of more reliable and powerful AI intelligent models is undoubtedly the unremitting pursuit by major banks in the financial industry. In this study, we used a dataset of over 40,000 records provided by a commercial bank as the research object. We compared various dimensionality reduction techniques such as PCA and T-SNE for preprocessing high-dimensional datasets and performed in-depth adaptation and tuning of distributed models such as LightGBM and XGBoost, as well as deep models like Tabnet. After a series of research and processing, we obtained excellent research results by combining SMOTEENN with these techniques. The experiments demonstrated that LightGBM combined with PCA and SMOTEENN techniques can assist banks in accurately predicting potential high-quality customers, showing relatively outstanding performance compared to other models.

Simulations are the best approximation to experimental laboratories in astrophysics and cosmology. However, the complexity, richness, and large size of their outputs severely limit the interpretability of their predictions. We describe a new, unbiased, and machine learning based approach to obtaining useful scientific insights from a broad range of simulations. The method can be used on today's largest simulations and will be essential to solve the extreme data exploration and analysis challenges posed by the Exascale era. Furthermore, this concept is so flexible, that it will also enable explorative access to observed data. Our concept is based on applying nonlinear dimensionality reduction to learn compact representations of the data in a low-dimensional space. The simulation data is projected onto this space for interactive inspection, visual interpretation, sample selection, and local analysis. We present a prototype using a rotational invariant hyperspherical variational convolutional autoencoder, utilizing a power distribution in the latent space, and trained on galaxies from IllustrisTNG simulation. Thereby, we obtain a natural Hubble tuning fork like similarity space that can be visualized interactively on the surface of a sphere by exploiting the power of HiPS tilings in Aladin Lite.

We present MiniTensor, an open source tensor operations library that focuses on minimalism, correctness, and performance. MiniTensor exposes a familiar PyTorch-like Python API while it executes performance critical code in a Rust engine. The core supports dense n dimensional tensors, broadcasting, reductions, matrix multiplication, reverse mode automatic differentiation, a compact set of neural network layers, and standard optimizers. In this paper, we describe the design of MiniTensor's architecture, including its efficient memory management, dynamic computation graph for gradients, and integration with Python via PyO3. We also compare the install footprint with PyTorch and TensorFlow to demonstrate that MiniTensor achieves a package size of only a few megabytes, several orders of magnitude smaller than mainstream frameworks, while preserving the essentials needed for research and development on CPUs. The repository can be found at https://github.com/neuralsorcerer/minitensor

Anticipating diverse future states is a central challenge in video world modeling. Discriminative world models produce a deterministic prediction that implicitly averages over possible futures, while existing generative world models remain computationally expensive. Recent work demonstrates that predicting the future in the feature space of a vision foundation model (VFM), rather than a latent space optimized for pixel reconstruction, requires significantly fewer world model parameters. However, most such approaches remain discriminative. In this work, we introduce DeltaTok, a tokenizer that encodes the VFM feature difference between consecutive frames into a single continuous "delta" token, and DeltaWorld, a generative world model operating on these tokens to efficiently generate diverse plausible futures. Delta tokens reduce video from a three-dimensional spatio-temporal representation to a one-dimensional temporal sequence, for example yielding a 1,024x token reduction with 512x512 frames. This compact representation enables tractable multi-hypothesis training, where many futures are generated in parallel and only the best is supervised. At inference, this leads to diverse predictions in a single forward pass. Experiments on dense forecasting tasks demonstrate that DeltaWorld forecasts futures that more closely align with real-world outcomes, while having over 35x fewer parameters and using 2,000x fewer FLOPs than existing generative world models. Code and weights: https://deltatok.github.io.

Multimodal recommendation systems are increasingly becoming foundational technologies for e-commerce and content platforms, enabling personalized services by jointly modeling users' historical behaviors and the multimodal features of items (e.g., visual and textual). However, most existing methods rely on either static fusion strategies or graph-based local interaction modeling, facing two critical limitations: (1) insufficient ability to model fine-grained cross-modal associations, leading to suboptimal fusion quality; and (2) a lack of global distribution-level consistency, causing representational bias. To address these, we propose MambaRec, a novel framework that integrates local feature alignment and global distribution regularization via attention-guided learning. At its core, we introduce the Dilated Refinement Attention Module (DREAM), which uses multi-scale dilated convolutions with channel-wise and spatial attention to align fine-grained semantic patterns between visual and textual modalities. This module captures hierarchical relationships and context-aware associations, improving cross-modal semantic modeling. Additionally, we apply Maximum Mean Discrepancy (MMD) and contrastive loss functions to constrain global modality alignment, enhancing semantic consistency. This dual regularization reduces mode-specific deviations and boosts robustness. To improve scalability, MambaRec employs a dimensionality reduction strategy to lower the computational cost of high-dimensional multimodal features. Extensive experiments on real-world e-commerce datasets show that MambaRec outperforms existing methods in fusion quality, generalization, and efficiency. Our code has been made publicly available at https://github.com/rkl71/MambaRec.

Many methods in differentially private model training rely on computing the similarity between a query point (such as public or synthetic data) and private data. We abstract out this common subroutine and study the following fundamental algorithmic problem: Given a similarity function f and a large high-dimensional private dataset X subset R^d, output a differentially private (DP) data structure which approximates sum_{x in X} f(x,y) for any query y. We consider the cases where f is a kernel function, such as f(x,y) = e^{-|x-y|_2^2/sigma^2} (also known as DP kernel density estimation), or a distance function such as f(x,y) = |x-y|_2, among others. Our theoretical results improve upon prior work and give better privacy-utility trade-offs as well as faster query times for a wide range of kernels and distance functions. The unifying approach behind our results is leveraging `low-dimensional structures' present in the specific functions f that we study, using tools such as provable dimensionality reduction, approximation theory, and one-dimensional decomposition of the functions. Our algorithms empirically exhibit improved query times and accuracy over prior state of the art. We also present an application to DP classification. Our experiments demonstrate that the simple methodology of classifying based on average similarity is orders of magnitude faster than prior DP-SGD based approaches for comparable accuracy.

Speech foundation models have significantly advanced various speech-related tasks by providing exceptional representation capabilities. However, their high-dimensional output features often create a mismatch with downstream task models, which typically require lower-dimensional inputs. A common solution is to apply a dimensionality reduction (DR) layer, but this approach increases parameter overhead, computational costs, and risks losing valuable information. To address these issues, we propose Nested Res2Net (Nes2Net), a lightweight back-end architecture designed to directly process high-dimensional features without DR layers. The nested structure enhances multi-scale feature extraction, improves feature interaction, and preserves high-dimensional information. We first validate Nes2Net on CtrSVDD, a singing voice deepfake detection dataset, and report a 22% performance improvement and an 87% back-end computational cost reduction over the state-of-the-art baseline. Additionally, extensive testing across four diverse datasets: ASVspoof 2021, ASVspoof 5, PartialSpoof, and In-the-Wild, covering fully spoofed speech, adversarial attacks, partial spoofing, and real-world scenarios, consistently highlights Nes2Net's superior robustness and generalization capabilities. The code package and pre-trained models are available at https://github.com/Liu-Tianchi/Nes2Net.

Modern deep learning models have the ability to generate high-dimensional vectors whose similarity reflects semantic resemblance. Thus, similarity search, i.e., the operation of retrieving those vectors in a large collection that are similar to a given query, has become a critical component of a wide range of applications that demand highly accurate and timely answers. In this setting, the high vector dimensionality puts similarity search systems under compute and memory pressure, leading to subpar performance. Additionally, cross-modal retrieval tasks have become increasingly common, e.g., where a user inputs a text query to find the most relevant images for that query. However, these queries often have different distributions than the database embeddings, making it challenging to achieve high accuracy. In this work, we present LeanVec, a framework that combines linear dimensionality reduction with vector quantization to accelerate similarity search on high-dimensional vectors while maintaining accuracy. We present LeanVec variants for in-distribution (ID) and out-of-distribution (OOD) queries. LeanVec-ID yields accuracies on par with those from recently introduced deep learning alternatives whose computational overhead precludes their usage in practice. LeanVec-OOD uses a novel technique for dimensionality reduction that considers the query and database distributions to simultaneously boost the accuracy and the performance of the framework even further (even presenting competitive results when the query and database distributions match). All in all, our extensive and varied experimental results show that LeanVec produces state-of-the-art results, with up to 3.7x improvement in search throughput and up to 4.9x faster index build time over the state of the art.

Recent advances in attention-free sequence models rely on convolutions as alternatives to the attention operator at the core of Transformers. In particular, long convolution sequence models have achieved state-of-the-art performance in many domains, but incur a significant cost during auto-regressive inference workloads -- naively requiring a full pass (or caching of activations) over the input sequence for each generated token -- similarly to attention-based models. In this paper, we seek to enable mathcal O(1) compute and memory cost per token in any pre-trained long convolution architecture to reduce memory footprint and increase throughput during generation. Concretely, our methods consist in extracting low-dimensional linear state-space models from each convolution layer, building upon rational interpolation and model-order reduction techniques. We further introduce architectural improvements to convolution-based layers such as Hyena: by weight-tying the filters across channels into heads, we achieve higher pre-training quality and reduce the number of filters to be distilled. The resulting model achieves 10x higher throughput than Transformers and 1.5x higher than Hyena at 1.3B parameters, without any loss in quality after distillation.

Optimal transport (OT) theory focuses, among all maps T:R^drightarrow R^d that can morph a probability measure onto another, on those that are the ``thriftiest'', i.e. such that the averaged cost c(x, T(x)) between x and its image T(x) be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when c is the ell_2^2 distance, e.g., using entropic maps [Pooladian'22], or neural networks [Makkuva'20, Korotin'20]. We propose a new model for transport maps, built on a family of translation invariant costs c(x, y):=h(x-y), where h:=1{2}|cdot|_2^2+tau and tau is a regularizer. We propose a generalization of the entropic map suitable for h, and highlight a surprising link tying it with the Bregman centroids of the divergence D_h generated by h, and the proximal operator of tau. We show that choosing a sparsity-inducing norm for tau results in maps that apply Occam's razor to transport, in the sense that the displacement vectors Delta(x):= T(x)-x they induce are sparse, with a sparsity pattern that varies depending on x. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data, in the 34000-d space of gene counts for cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.

In this paper, we present DendroMap, a novel approach to interactively exploring large-scale image datasets for machine learning (ML). ML practitioners often explore image datasets by generating a grid of images or projecting high-dimensional representations of images into 2-D using dimensionality reduction techniques (e.g., t-SNE). However, neither approach effectively scales to large datasets because images are ineffectively organized and interactions are insufficiently supported. To address these challenges, we develop DendroMap by adapting Treemaps, a well-known visualization technique. DendroMap effectively organizes images by extracting hierarchical cluster structures from high-dimensional representations of images. It enables users to make sense of the overall distributions of datasets and interactively zoom into specific areas of interests at multiple levels of abstraction. Our case studies with widely-used image datasets for deep learning demonstrate that users can discover insights about datasets and trained models by examining the diversity of images, identifying underperforming subgroups, and analyzing classification errors. We conducted a user study that evaluates the effectiveness of DendroMap in grouping and searching tasks by comparing it with a gridified version of t-SNE and found that participants preferred DendroMap. DendroMap is available at https://div-lab.github.io/dendromap/.

Extracting insight from the enormous quantity of data generated from molecular simulations requires the identification of a small number of collective variables whose corresponding low-dimensional free-energy landscape retains the essential features of the underlying system. Data-driven techniques provide a systematic route to constructing this landscape, without the need for extensive a priori intuition into the relevant driving forces. In particular, autoencoders are powerful tools for dimensionality reduction, as they naturally force an information bottleneck and, thereby, a low-dimensional embedding of the essential features. While variational autoencoders ensure continuity of the embedding by assuming a unimodal Gaussian prior, this is at odds with the multi-basin free-energy landscapes that typically arise from the identification of meaningful collective variables. In this work, we incorporate this physical intuition into the prior by employing a Gaussian mixture variational autoencoder (GMVAE), which encourages the separation of metastable states within the embedding. The GMVAE performs dimensionality reduction and clustering within a single unified framework, and is capable of identifying the inherent dimensionality of the input data, in terms of the number of Gaussians required to categorize the data. We illustrate our approach on two toy models, alanine dipeptide, and a challenging disordered peptide ensemble, demonstrating the enhanced clustering effect of the GMVAE prior compared to standard VAEs. The resulting embeddings appear to be promising representations for constructing Markov state models, highlighting the transferability of the dimensionality reduction from static equilibrium properties to dynamics.

Cognitive neuroscience is enjoying rapid increase in extensive public brain-imaging datasets. It opens the door to large-scale statistical models. Finding a unified perspective for all available data calls for scalable and automated solutions to an old challenge: how to aggregate heterogeneous information on brain function into a universal cognitive system that relates mental operations/cognitive processes/psychological tasks to brain networks? We cast this challenge in a machine-learning approach to predict conditions from statistical brain maps across different studies. For this, we leverage multi-task learning and multi-scale dimension reduction to learn low-dimensional representations of brain images that carry cognitive information and can be robustly associated with psychological stimuli. Our multi-dataset classification model achieves the best prediction performance on several large reference datasets, compared to models without cognitive-aware low-dimension representations, it brings a substantial performance boost to the analysis of small datasets, and can be introspected to identify universal template cognitive concepts.

Low-Rank Adaptation (LoRA) has become the de facto parameter-efficient fine-tuning (PEFT) method for large language models (LLMs) by constraining weight updates to low-rank matrices. Recent works such as Tied-LoRA, VeRA, and VB-LoRA push efficiency further by introducing additional constraints to reduce the trainable parameter space. In this paper, we show that the parameter space reduction strategies employed by these LoRA variants can be formulated within a unified framework, Uni-LoRA, where the LoRA parameter space, flattened as a high-dimensional vector space R^D, can be reconstructed through a projection from a subspace R^d, with d ll D. We demonstrate that the fundamental difference among various LoRA methods lies in the choice of the projection matrix, P in R^{D times d}.Most existing LoRA variants rely on layer-wise or structure-specific projections that limit cross-layer parameter sharing, thereby compromising parameter efficiency. In light of this, we introduce an efficient and theoretically grounded projection matrix that is isometric, enabling global parameter sharing and reducing computation overhead. Furthermore, under the unified view of Uni-LoRA, this design requires only a single trainable vector to reconstruct LoRA parameters for the entire LLM - making Uni-LoRA both a unified framework and a "one-vector-only" solution. Extensive experiments on GLUE, mathematical reasoning, and instruction tuning benchmarks demonstrate that Uni-LoRA achieves state-of-the-art parameter efficiency while outperforming or matching prior approaches in predictive performance. Our code is available at https://github.com/KaiyangLi1992/Uni-LoRA.

Structured pruning is a popular method to reduce the cost of convolutional neural networks, that are the state of the art in many computer vision tasks. However, depending on the architecture, pruning introduces dimensional discrepancies which prevent the actual reduction of pruned networks. To tackle this problem, we propose a method that is able to take any structured pruning mask and generate a network that does not encounter any of these problems and can be leveraged efficiently. We provide an accurate description of our solution and show results of gains, in energy consumption and inference time on embedded hardware, of pruned convolutional neural networks.

Recent work has shown that diffusion models can generate high-quality images by operating directly on SSL patch features rather than pixel-space latents. However, the dense patch grids from encoders like DINOv2 contain significant redundancy, making diffusion needlessly expensive. We introduce FlatDINO, a variational autoencoder that compresses this representation into a one-dimensional sequence of just 32 continuous tokens -an 8x reduction in sequence length and 48x compression in total dimensionality. On ImageNet 256x256, a DiT-XL trained on FlatDINO latents achieves a gFID of 1.80 with classifier-free guidance while requiring 8x fewer FLOPs per forward pass and up to 4.5x fewer FLOPs per training step compared to diffusion on uncompressed DINOv2 features. These are preliminary results and this work is in progress.

In meta reinforcement learning (meta RL), an agent learns from a set of training tasks how to quickly solve a new task, drawn from the same task distribution. The optimal meta RL policy, a.k.a. the Bayes-optimal behavior, is well defined, and guarantees optimal reward in expectation, taken with respect to the task distribution. The question we explore in this work is how many training tasks are required to guarantee approximately optimal behavior with high probability. Recent work provided the first such PAC analysis for a model-free setting, where a history-dependent policy was learned from the training tasks. In this work, we propose a different approach: directly learn the task distribution, using density estimation techniques, and then train a policy on the learned task distribution. We show that our approach leads to bounds that depend on the dimension of the task distribution. In particular, in settings where the task distribution lies in a low-dimensional manifold, we extend our analysis to use dimensionality reduction techniques and account for such structure, obtaining significantly better bounds than previous work, which strictly depend on the number of states and actions. The key of our approach is the regularization implied by the kernel density estimation method. We further demonstrate that this regularization is useful in practice, when `plugged in' the state-of-the-art VariBAD meta RL algorithm.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to O(d^{k}) scaling of the derivative tensor size and the O(2^{k-1}L) scaling in the computation graph, where d is the dimension of the domain, L is the number of ops in the forward computation graph, and k is the derivative order. In previous works, the polynomial scaling in d was addressed by amortizing the computation over the optimization process via randomization. Separately, the exponential scaling in k for univariate functions (d=1) was addressed with high-order auto-differentiation (AD). In this work, we show how to efficiently perform arbitrary contraction of the derivative tensor of arbitrary order for multivariate functions, by properly constructing the input tangents to univariate high-order AD, which can be used to efficiently randomize any differential operator. When applied to Physics-Informed Neural Networks (PINNs), our method provides >1000times speed-up and >30times memory reduction over randomization with first-order AD, and we can now solve 1-million-dimensional PDEs in 8 minutes on a single NVIDIA A100 GPU. This work opens the possibility of using high-order differential operators in large-scale problems.

Brain-computer interfaces (BCIs) offer transformative potential, but decoding neural signals presents significant challenges. The core premise of this paper is built around demonstrating methods to elucidate the underlying low-dimensional geometric structure present in high-dimensional brainwave data in order to assist in downstream BCI-related neural classification tasks. We demonstrate two pipelines related to electroencephalography (EEG) signal processing: (1) a preliminary pipeline removing noise from individual EEG channels, and (2) a downstream manifold learning pipeline uncovering geometric structure across networks of EEG channels. We conduct preliminary validation using two EEG datasets and situate our demonstration in the context of the BCI-relevant imagined digit decoding problem. Our preliminary pipeline uses an attention-based EEG filtration network to extract clean signal from individual EEG channels. Our primary pipeline uses a fast Fourier transform, a Laplacian eigenmap, a discrete analog of Ricci flow via Ollivier's notion of Ricci curvature, and a graph convolutional network to perform dimensionality reduction on high-dimensional multi-channel EEG data in order to enable regularizable downstream classification. Our system achieves competitive performance with existing signal processing and classification benchmarks; we demonstrate a mean test correlation coefficient of >0.95 at 2 dB on semi-synthetic neural denoising and a downstream EEG-based classification accuracy of 0.97 on distinguishing digit- versus non-digit- thoughts. Results are preliminary and our geometric machine learning pipeline should be validated by more extensive follow-up studies; generalizing these results to larger inter-subject sample sizes, different hardware systems, and broader use cases will be crucial.

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

We propose Super-resolution Neural Operator (SRNO), a deep operator learning framework that can resolve high-resolution (HR) images at arbitrary scales from the low-resolution (LR) counterparts. Treating the LR-HR image pairs as continuous functions approximated with different grid sizes, SRNO learns the mapping between the corresponding function spaces. From the perspective of approximation theory, SRNO first embeds the LR input into a higher-dimensional latent representation space, trying to capture sufficient basis functions, and then iteratively approximates the implicit image function with a kernel integral mechanism, followed by a final dimensionality reduction step to generate the RGB representation at the target coordinates. The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to "hallucinate" high-frequency information from the LR image. Experiments show that SRNO outperforms existing continuous SR methods in terms of both accuracy and running time. Our code is at https://github.com/2y7c3/Super-Resolution-Neural-Operator

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

State-of-the-art deep learning models have a parameter count that reaches into the billions. Training, storing and transferring such models is energy and time consuming, thus costly. A big part of these costs is caused by training the network. Model compression lowers storage and transfer costs, and can further make training more efficient by decreasing the number of computations in the forward and/or backward pass. Thus, compressing networks also at training time while maintaining a high performance is an important research topic. This work is a survey on methods which reduce the number of trained weights in deep learning models throughout the training. Most of the introduced methods set network parameters to zero which is called pruning. The presented pruning approaches are categorized into pruning at initialization, lottery tickets and dynamic sparse training. Moreover, we discuss methods that freeze parts of a network at its random initialization. By freezing weights, the number of trainable parameters is shrunken which reduces gradient computations and the dimensionality of the model's optimization space. In this survey we first propose dimensionality reduced training as an underlying mathematical model that covers pruning and freezing during training. Afterwards, we present and discuss different dimensionality reduced training methods.

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

Neural multivariate regression underpins a wide range of domains such as control, robotics, and finance, yet the geometry of its learned representations remains poorly characterized. While neural collapse has been shown to benefit generalization in classification, we find that analogous collapse in regression consistently degrades performance. To explain this contrast, we analyze models through the lens of intrinsic dimension. Across control tasks and synthetic datasets, we estimate the intrinsic dimension of last-layer features (ID_H) and compare it with that of the regression targets (ID_Y). Collapsed models exhibit ID_H < ID_Y, leading to over-compression and poor generalization, whereas non-collapsed models typically maintain ID_H > ID_Y. For the non-collapsed models, performance with respect to ID_H depends on the data quantity and noise levels. From these observations, we identify two regimes (over-compressed and under-compressed) that determine when expanding or reducing feature dimensionality improves performance. Our results provide new geometric insights into neural regression and suggest practical strategies for enhancing generalization.

Motor impairments, frequently caused by neurological incidents like strokes or traumatic brain injuries, present substantial obstacles in rehabilitation therapy. This research aims to elevate the field by optimizing motor imagery classification algorithms within Brain-Computer Interfaces (BCIs). By improving the efficiency of BCIs, we offer a novel approach that holds significant promise for enhancing motor rehabilitation outcomes. Utilizing unsupervised techniques for dimensionality reduction, namely Uniform Manifold Approximation and Projection (UMAP) coupled with K-Nearest Neighbors (KNN), we evaluate the necessity of employing supervised methods such as Long Short-Term Memory (LSTM) and Convolutional Neural Networks (CNNs) for classification tasks. Importantly, participants who exhibited high KNN scores following UMAP dimensionality reduction also achieved high accuracy in supervised deep learning (DL) models. Due to individualized model requirements and massive neural training data, dimensionality reduction becomes an effective preprocessing step that minimizes the need for extensive data labeling and supervised deep learning techniques. This approach has significant implications not only for targeted therapies in motor dysfunction but also for addressing regulatory, safety, and reliability concerns in the rapidly evolving BCI field.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

Currently, high-dimensional data is ubiquitous in data science, which necessitates the development of techniques to decompose and interpret such multidimensional (aka tensor) datasets. Finding a low dimensional representation of the data, that is, its inherent structure, is one of the approaches that can serve to understand the dynamics of low dimensional latent features hidden in the data. Nonnegative RESCAL is one such technique, particularly well suited to analyze self-relational data, such as dynamic networks found in international trade flows. Nonnegative RESCAL computes a low dimensional tensor representation by finding the latent space containing multiple modalities. Estimating the dimensionality of this latent space is crucial for extracting meaningful latent features. Here, to determine the dimensionality of the latent space with nonnegative RESCAL, we propose a latent dimension determination method which is based on clustering of the solutions of multiple realizations of nonnegative RESCAL decompositions. We demonstrate the performance of our model selection method on synthetic data and then we apply our method to decompose a network of international trade flows data from International Monetary Fund and validate the resulting features against empirical facts from economic literature.

The manifold hypothesis posits that high-dimensional data often lies on a lower-dimensional manifold and that utilizing this manifold as the target space yields more efficient representations. While numerous traditional manifold-based techniques exist for dimensionality reduction, their application in self-supervised learning has witnessed slow progress. The recent MSimCLR method combines manifold encoding with SimCLR but requires extremely low target encoding dimensions to outperform SimCLR, limiting its applicability. This paper introduces a novel learning paradigm using an unbalanced atlas (UA), capable of surpassing state-of-the-art self-supervised learning approaches. We investigated and engineered the DeepInfomax with an unbalanced atlas (DIM-UA) method by adapting the Spatiotemporal DeepInfomax (ST-DIM) framework to align with our proposed UA paradigm. The efficacy of DIM-UA is demonstrated through training and evaluation on the Atari Annotated RAM Interface (AtariARI) benchmark, a modified version of the Atari 2600 framework that produces annotated image samples for representation learning. The UA paradigm improves existing algorithms significantly as the number of target encoding dimensions grows. For instance, the mean F1 score averaged over categories of DIM-UA is ~75% compared to ~70% of ST-DIM when using 16384 hidden units.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

We study the problem of visualizing large-scale and high-dimensional data in a low-dimensional (typically 2D or 3D) space. Much success has been reported recently by techniques that first compute a similarity structure of the data points and then project them into a low-dimensional space with the structure preserved. These two steps suffer from considerable computational costs, preventing the state-of-the-art methods such as the t-SNE from scaling to large-scale and high-dimensional data (e.g., millions of data points and hundreds of dimensions). We propose the LargeVis, a technique that first constructs an accurately approximated K-nearest neighbor graph from the data and then layouts the graph in the low-dimensional space. Comparing to t-SNE, LargeVis significantly reduces the computational cost of the graph construction step and employs a principled probabilistic model for the visualization step, the objective of which can be effectively optimized through asynchronous stochastic gradient descent with a linear time complexity. The whole procedure thus easily scales to millions of high-dimensional data points. Experimental results on real-world data sets demonstrate that the LargeVis outperforms the state-of-the-art methods in both efficiency and effectiveness. The hyper-parameters of LargeVis are also much more stable over different data sets.

Dimensionality reduction in vector databases is pivotal for streamlining AI data management, enabling efficient storage, faster computation, and improved model performance. This paper explores the benefits of reducing vector database dimensions, with a focus on computational efficiency and overcoming the curse of dimensionality. We introduce a novel application of Fast Fourier Transform (FFT) to dimensionality reduction, a method previously underexploited in this context. By demonstrating its utility across various AI domains, including Retrieval-Augmented Generation (RAG) models and image processing, this FFT-based approach promises to improve data retrieval processes and enhance the efficiency and scalability of AI solutions. The incorporation of FFT may not only optimize operations in real-time processing and recommendation systems but also extend to advanced image processing techniques, where dimensionality reduction can significantly improve performance and analysis efficiency. This paper advocates for the broader adoption of FFT in vector database management, marking a significant stride towards addressing the challenges of data volume and complexity in AI research and applications. Unlike many existing approaches, we directly handle the embedding vectors produced by the model after processing a test input.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

Diffusion models, which learn to reverse a signal destruction process to generate new data, typically require the signal at each step to have the same dimension. We argue that, considering the spatial redundancy in image signals, there is no need to maintain a high dimensionality in the evolution process, especially in the early generation phase. To this end, we make a theoretical generalization of the forward diffusion process via signal decomposition. Concretely, we manage to decompose an image into multiple orthogonal components and control the attenuation of each component when perturbing the image. That way, along with the noise strength increasing, we are able to diminish those inconsequential components and thus use a lower-dimensional signal to represent the source, barely losing information. Such a reformulation allows to vary dimensions in both training and inference of diffusion models. Extensive experiments on a range of datasets suggest that our approach substantially reduces the computational cost and achieves on-par or even better synthesis performance compared to baseline methods. We also show that our strategy facilitates high-resolution image synthesis and improves FID of diffusion model trained on FFHQ at 1024times1024 resolution from 52.40 to 10.46. Code and models will be made publicly available.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

Although pretrained language models can be fine-tuned to produce state-of-the-art results for a very wide range of language understanding tasks, the dynamics of this process are not well understood, especially in the low data regime. Why can we use relatively vanilla gradient descent algorithms (e.g., without strong regularization) to tune a model with hundreds of millions of parameters on datasets with only hundreds or thousands of labeled examples? In this paper, we argue that analyzing fine-tuning through the lens of intrinsic dimension provides us with empirical and theoretical intuitions to explain this remarkable phenomenon. We empirically show that common pre-trained models have a very low intrinsic dimension; in other words, there exists a low dimension reparameterization that is as effective for fine-tuning as the full parameter space. For example, by optimizing only 200 trainable parameters randomly projected back into the full space, we can tune a RoBERTa model to achieve 90\% of the full parameter performance levels on MRPC. Furthermore, we empirically show that pre-training implicitly minimizes intrinsic dimension and, perhaps surprisingly, larger models tend to have lower intrinsic dimension after a fixed number of pre-training updates, at least in part explaining their extreme effectiveness. Lastly, we connect intrinsic dimensionality with low dimensional task representations and compression based generalization bounds to provide intrinsic-dimension-based generalization bounds that are independent of the full parameter count.

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

Retrieval-Augmented Generation (RAG) has emerged as a powerful paradigm for grounding large language models in external knowledge sources, improving the precision of agents responses. However, high-dimensional language model embeddings, often in the range of hundreds to thousands of dimensions, can present scalability challenges in terms of storage and latency, especially when processing massive financial text corpora. This paper investigates the use of Principal Component Analysis (PCA) to reduce embedding dimensionality, thereby mitigating computational bottlenecks without incurring large accuracy losses. We experiment with a real-world dataset and compare different similarity and distance metrics under both full-dimensional and PCA-compressed embeddings. Our results show that reducing vectors from 3,072 to 110 dimensions provides a sizeable (up to 60times) speedup in retrieval operations and a sim 28.6times reduction in index size, with only moderate declines in correlation metrics relative to human-annotated similarity scores. These findings demonstrate that PCA-based compression offers a viable balance between retrieval fidelity and resource efficiency, essential for real-time systems such as Zanista AI's Newswitch platform. Ultimately, our study underscores the practicality of leveraging classical dimensionality reduction techniques to scale RAG architectures for knowledge-intensive applications in finance and trading, where speed, memory efficiency, and accuracy must jointly be optimized.

In high dimension, low sample size (HDLSS) settings, classifiers based on Euclidean distances like the nearest neighbor classifier and the average distance classifier perform quite poorly if differences between locations of the underlying populations get masked by scale differences. To rectify this problem, several modifications of these classifiers have been proposed in the literature. However, existing methods are confined to location and scale differences only, and often fail to discriminate among populations differing outside of the first two moments. In this article, we propose some simple transformations of these classifiers resulting into improved performance even when the underlying populations have the same location and scale. We further propose a generalization of these classifiers based on the idea of grouping of variables. The high-dimensional behavior of the proposed classifiers is studied theoretically. Numerical experiments with a variety of simulated examples as well as an extensive analysis of real data sets exhibit advantages of the proposed methods.

Vision Transformer (ViT) extends the application range of transformers from language processing to computer vision tasks as being an alternative architecture against the existing convolutional neural networks (CNN). Since the transformer-based architecture has been innovative for computer vision modeling, the design convention towards an effective architecture has been less studied yet. From the successful design principles of CNN, we investigate the role of spatial dimension conversion and its effectiveness on transformer-based architecture. We particularly attend to the dimension reduction principle of CNNs; as the depth increases, a conventional CNN increases channel dimension and decreases spatial dimensions. We empirically show that such a spatial dimension reduction is beneficial to a transformer architecture as well, and propose a novel Pooling-based Vision Transformer (PiT) upon the original ViT model. We show that PiT achieves the improved model capability and generalization performance against ViT. Throughout the extensive experiments, we further show PiT outperforms the baseline on several tasks such as image classification, object detection, and robustness evaluation. Source codes and ImageNet models are available at https://github.com/naver-ai/pit

Practitioners prune neural networks for efficiency gains and generalization improvements, but few scrutinize the factors determining the prunability of a neural network the maximum fraction of weights that pruning can remove without compromising the model's test accuracy. In this work, we study the properties of input data that may contribute to the prunability of a neural network. For high dimensional input data such as images, text, and audio, the manifold hypothesis suggests that these high dimensional inputs approximately lie on or near a significantly lower dimensional manifold. Prior work demonstrates that the underlying low dimensional structure of the input data may affect the sample efficiency of learning. In this paper, we investigate whether the low dimensional structure of the input data affects the prunability of a neural network.

Extracting low-dimensional summary statistics from large datasets is essential for efficient (likelihood-free) inference. We characterize different classes of summaries and demonstrate their importance for correctly analysing dimensionality reduction algorithms. We demonstrate that minimizing the expected posterior entropy (EPE) under the prior predictive distribution of the model subsumes many existing methods. They are equivalent to or are special or limiting cases of minimizing the EPE. We offer a unifying framework for obtaining informative summaries, provide concrete recommendations for practitioners, and propose a practical method to obtain high-fidelity summaries whose utility we demonstrate for both benchmark and practical examples.

Modern recommender systems struggle to effectively utilize the rich, yet high-dimensional and noisy, multi-modal features generated by Large Language Models (LLMs). Treating these features as static inputs decouples them from the core recommendation task. We address this limitation with a novel framework built on a key insight: deeply fusing multi-modal and collaborative knowledge for representation denoising. Our unified architecture introduces two primary technical innovations. First, we integrate dimensionality reduction directly into the recommendation model, enabling end-to-end co-training that makes the reduction process aware of the final ranking objective. Second, we introduce a contrastive learning objective that explicitly incorporates the collaborative filtering signal into the latent space. This synergistic process refines raw LLM embeddings, filtering noise while amplifying task-relevant signals. Extensive experiments confirm our method's superior discriminative power, proving that this integrated fusion and denoising strategy is critical for achieving state-of-the-art performance. Our work provides a foundational paradigm for effectively harnessing LLMs in recommender systems.

The article explores an encoding and structural information processing approach using sparse bit vectors and fixed-length linear vectors. The following are presented: a discrete method of speculative stochastic dimensionality reduction of multidimensional code and linear spaces with linear asymptotic complexity; a geometric method for obtaining discrete embeddings of an organised code space that reflect the internal structure of a given modality. The structure and properties of a code space are investigated using three modalities as examples: morphology of Russian and English languages, and immunohistochemical markers. Parallels are drawn between the resulting map of the code space layout and so-called pinwheels appearing on the mammalian neocortex. A cautious assumption is made about similarities between neocortex organisation and processes happening in our models.

Drawing motivation from the manifold hypothesis, which posits that most high-dimensional data lies on or near low-dimensional manifolds, we apply manifold learning to the space of neural networks. We learn manifolds where datapoints are neural networks by introducing a distance between the hidden layer representations of the neural networks. These distances are then fed to the non-linear dimensionality reduction algorithm PHATE to create a manifold of neural networks. We characterize this manifold using features of the representation, including class separation, hierarchical cluster structure, spectral entropy, and topological structure. Our analysis reveals that high-performing networks cluster together in the manifold, displaying consistent embedding patterns across all these features. Finally, we demonstrate the utility of this approach for guiding hyperparameter optimization and neural architecture search by sampling from the manifold.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

In this study, we present an investigation into the anisotropy dynamics and intrinsic dimension of embeddings in transformer architectures, focusing on the dichotomy between encoders and decoders. Our findings reveal that the anisotropy profile in transformer decoders exhibits a distinct bell-shaped curve, with the highest anisotropy concentrations in the middle layers. This pattern diverges from the more uniformly distributed anisotropy observed in encoders. In addition, we found that the intrinsic dimension of embeddings increases in the initial phases of training, indicating an expansion into higher-dimensional space. Which is then followed by a compression phase towards the end of training with dimensionality decrease, suggesting a refinement into more compact representations. Our results provide fresh insights to the understanding of encoders and decoders embedding properties.

We investigate the geometric structure of learning dynamics in overparameterized transformer models through carefully controlled modular arithmetic tasks. Our primary finding is that despite operating in high-dimensional parameter spaces (d=128), transformer training trajectories rapidly collapse onto low-dimensional execution manifolds of dimension 3--4. This dimensional collapse is robust across random seeds and moderate task difficulties, though the orientation of the manifold in parameter space varies between runs. We demonstrate that this geometric structure underlies several empirically observed phenomena: (1) sharp attention concentration emerges as saturation along routing coordinates within the execution manifold, (2) SGD commutators are preferentially aligned with the execution subspace (up to 10times random baseline) early in training, with >92% of non-commutativity confined to orthogonal staging directions and this alignment decreasing as training converges, and (3) sparse autoencoders capture auxiliary routing structure but fail to isolate execution itself, which remains distributed across the low-dimensional manifold. Our results suggest a unifying geometric framework for understanding transformer learning, where the vast majority of parameters serve to absorb optimization interference while core computation occurs in a dramatically reduced subspace. These findings have implications for interpretability, training curriculum design, and understanding the role of overparameterization in neural network learning.

We tackle the problem disentangling the latent space of an autoencoder in order to separate labelled attribute information from other characteristic information. This then allows us to change selected attributes while preserving other information. Our method, matrix subspace projection, is much simpler than previous approaches to latent space factorisation, for example not requiring multiple discriminators or a careful weighting among their loss functions. Furthermore our new model can be applied to autoencoders as a plugin, and works across diverse domains such as images or text. We demonstrate the utility of our method for attribute manipulation in autoencoders trained across varied domains, using both human evaluation and automated methods. The quality of generation of our new model (e.g. reconstruction, conditional generation) is highly competitive to a number of strong baselines.

Deep neural networks perform remarkably well on image classification tasks but remain vulnerable to carefully crafted adversarial perturbations. This work revisits linear dimensionality reduction as a simple, data-adapted defense. We empirically compare standard Principal Component Analysis (PCA) with its sparse variant (SPCA) as front-end feature extractors for downstream classifiers, and we complement these experiments with a theoretical analysis. On the theory side, we derive exact robustness certificates for linear heads applied to SPCA features: for both ell_infty and ell_2 threat models (binary and multiclass), the certified radius grows as the dual norms of W^top u shrink, where W is the projection and u the head weights. We further show that for general (non-linear) heads, sparsity reduces operator-norm bounds through a Lipschitz composition argument, predicting lower input sensitivity. Empirically, with a small non-linear network after the projection, SPCA consistently degrades more gracefully than PCA under strong white-box and black-box attacks while maintaining competitive clean accuracy. Taken together, the theory identifies the mechanism (sparser projections reduce adversarial leverage) and the experiments verify that this benefit persists beyond the linear setting. Our code is available at https://github.com/killian31/SPCARobustness.

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.

Chain-of-thought (CoT) reasoning and its variants have substantially improved the performance of language models on complex reasoning tasks, yet the precise mechanisms by which different strategies facilitate generalization remain poorly understood. While current explanations often point to increased test-time computation or structural guidance, establishing a consistent, quantifiable link between these factors and generalization remains challenging. In this work, we identify intrinsic dimensionality as a quantitative measure for characterizing the effectiveness of reasoning chains. Intrinsic dimensionality quantifies the minimum number of model dimensions needed to reach a given accuracy threshold on a given task. By keeping the model architecture fixed and varying the task formulation through different reasoning strategies, we demonstrate that effective reasoning strategies consistently reduce the intrinsic dimensionality of the task. Validating this on GSM8K with Gemma-3 1B and 4B, we observe a strong inverse correlation between the intrinsic dimensionality of a reasoning strategy and its generalization performance on both in-distribution and out-of-distribution data. Our findings suggest that effective reasoning chains facilitate learning by better compressing the task using fewer parameters, offering a new quantitative metric for analyzing reasoning processes.

Classification algorithms have recently found applications in computational physics for the selection of numerical methods or models adapted to the environment and the state of the physical system. For such classification tasks, labeled training data come from numerical simulations and generally correspond to physical fields discretized on a mesh. Three challenging difficulties arise: the lack of training data, their high dimensionality, and the non-applicability of common data augmentation techniques to physics data. This article introduces two algorithms to address these issues, one for dimensionality reduction via feature selection, and one for data augmentation. These algorithms are combined with a wide variety of classifiers for their evaluation. When combined with a stacking ensemble made of six multilayer perceptrons and a ridge logistic regression, they enable reaching an accuracy of 90% on our classification problem for nonlinear structural mechanics.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

Supervised machine learning methods for geological mapping via remote sensing face limitations due to the scarcity of accurately labelled training data that can be addressed by unsupervised learning, such as dimensionality reduction and clustering. Dimensionality reduction methods have the potential to play a crucial role in improving the accuracy of geological maps. Although conventional dimensionality reduction methods may struggle with nonlinear data, unsupervised deep learning models such as autoencoders can model non-linear relationships. Stacked autoencoders feature multiple interconnected layers to capture hierarchical data representations useful for remote sensing data. We present an unsupervised machine learning-based framework for processing remote sensing data using stacked autoencoders for dimensionality reduction and k-means clustering for mapping geological units. We use Landsat 8, ASTER, and Sentinel-2 datasets to evaluate the framework for geological mapping of the Mutawintji region in Western New South Wales, Australia. We also compare stacked autoencoders with principal component analysis (PCA) and canonical autoencoders. Our results reveal that the framework produces accurate and interpretable geological maps, efficiently discriminating rock units. The results reveal that the combination of stacked autoencoders with Sentinel-2 data yields the best performance accuracy when compared to other combinations. We find that stacked autoencoders enable better extraction of complex and hierarchical representations of the input data when compared to canonical autoencoders and PCA. We also find that the generated maps align with prior geological knowledge of the study area while providing novel insights into geological structures.

Model dimension (d_{model}) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored. Grounded in the Linear Representation and Superposition Hypotheses - which propose that models encode features as near-orthogonal directions in latent space - we develop a framework for estimating how many such directions a model can support. We first establish the embedding matrix as a measurable proxy for near-orthogonality constraints across the latent space: the boundary between meaningful token relationships and incidental similarity in the pairwise cosine similarity distribution gives a concrete estimate of the model's accepted deviation varepsilon from perfect orthogonality. Applying this metric across dozens of open-source models reveals two classes: models with high varepsilon whose embeddings lack near-orthogonal structure, and models with low varepsilon that maintain it. We then show that the standard Johnson-Lindenstrauss lemma greatly underestimates the packing efficiency of trained representations, and derive an adjusted capacity formula in which the number of near-orthogonal directions depends on the ratio of vectors to dimensions (k/d) rather than the raw count - a single modification that cuts prediction error by two orders of magnitude with no extra parameters. Combining these results, we define representational capacity as an upper bound on the number of distinguishable directions available for features and embeddings in a model's latent space. Capacity is exponentially sensitive to varepsilon, and larger models favor tighter orthogonality constraints over maximizing raw capacity - a pattern compatible with several explanations (a stability-capacity trade-off, a ceiling on usable concepts, or confounds with model scale) that we leave to future work.

Why does the low dimensionality of representations, typically dapprox 1000, not prevent modern embedding-based retrieval models from scaling to billions, or even trillions, of data points? To answer this question, we study maximal-margin embeddings in the following retrieval model, classically studied in communication complexity [PS86] and more recently in embedding-based retrieval [WBNL26]. Let Ain {0,1}^{Ntimes n} be a matrix indicating whether each of N queries is relevant to each of n documents. We are interested in the largest margin m>0, denoted by m^{rd}(d, A), for which there exist unit norm embeddings of the queries and documents {U_j}_{j = 1}^N, {V_i}_{i = 1}^n with the following property. langle U_j, V_irangle ge m whenever A_{ji} = 1 and langle U_j, V_irangle le -m otherwise. A large margin is a key proxy for representation quality: it controls both robustness to perturbations and compositional generalization across queries. Our main theorem establishes that the best possible margin without a restriction on the dimension, m^{rd}(+infty, A), can be nearly achieved in dimension d = O(m^{rd}(+infty, A)^{-2}log n) which improves a theorem of [BDES02]. Together with a matching lower bound in Theorem 1.5, we conclude that when Ain {0,1}^{n{k}times n} is the matrix containing all possible k-sparse rows once, dimension d = O(klog (n/k)) is necessary and sufficient for the maximal possible margin m^{rd}(+infty, A) = Θ(k^{-1/2}) in this setting. This fully resolves the setup of [WBNL26]. We also give several constructions for large margins when d = o(klog (n/k)). Finally, we empirically test the InfoNCE and sigmoid losses for producing large margin embeddings and demonstrate a clear advantage of the sigmoid loss.

Recent works in robotic manipulation through reinforcement learning (RL) or imitation learning (IL) have shown potential for tackling a range of tasks e.g., opening a drawer or a cupboard. However, these techniques generalize poorly to unseen objects. We conjecture that this is due to the high-dimensional action space for joint control. In this paper, we take an alternative approach and separate the task of learning 'what to do' from 'how to do it' i.e., whole-body control. We pose the RL problem as one of determining the skill dynamics for a disembodied virtual manipulator interacting with articulated objects. The whole-body robotic kinematic control is optimized to execute the high-dimensional joint motion to reach the goals in the workspace. It does so by solving a quadratic programming (QP) model with robotic singularity and kinematic constraints. Our experiments on manipulating complex articulated objects show that the proposed approach is more generalizable to unseen objects with large intra-class variations, outperforming previous approaches. The evaluation results indicate that our approach generates more compliant robotic motion and outperforms the pure RL and IL baselines in task success rates. Additional information and videos are available at https://kl-research.github.io/decoupskill

Deep Neural Networks (DNNs) have been a large driver and enabler for AI breakthroughs in recent years. These models have been getting larger in their attempt to become more accurate and tackle new upcoming use-cases, including AR/VR and intelligent assistants. However, the training process of such large models is a costly and time-consuming process, which typically yields a single model to fit all targets. To mitigate this, various techniques have been proposed in the literature, including pruning, sparsification or quantization of the model weights and updates. While able to achieve high compression rates, they often incur computational overheads or accuracy penalties. Alternatively, factorization methods have been leveraged to incorporate low-rank compression in the training process. Similarly, such techniques (e.g.,~SVD) frequently rely on the computationally expensive decomposition of layers and are potentially sub-optimal for non-linear models, such as DNNs. In this work, we take a further step in designing efficient low-rank models and propose Maestro, a framework for trainable low-rank layers. Instead of regularly applying a priori decompositions such as SVD, the low-rank structure is built into the training process through a generalized variant of Ordered Dropout. This method imposes an importance ordering via sampling on the decomposed DNN structure. Our theoretical analysis demonstrates that our method recovers the SVD decomposition of linear mapping on uniformly distributed data and PCA for linear autoencoders. We further apply our technique on DNNs and empirically illustrate that Maestro enables the extraction of lower footprint models that preserve model performance while allowing for graceful accuracy-latency tradeoff for the deployment to devices of different capabilities.

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in 100,000 effective dimensions in 12 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

We investigate the relationship between representation geometry and neural network performance. Analyzing 52 pretrained ImageNet models across 13 architecture families, we show that effective dimension -- an unsupervised geometric metric -- strongly predicts accuracy. Output effective dimension achieves partial r=0.75 (p < 10^(-10)) after controlling for model capacity, while total compression achieves partial r=-0.72. These findings replicate across ImageNet and CIFAR-10, and generalize to NLP: effective dimension predicts performance for 8 encoder models on SST-2/MNLI and 15 decoder-only LLMs on AG News (r=0.69, p=0.004), while model size does not (r=0.07). We establish bidirectional causality: degrading geometry via noise causes accuracy loss (r=-0.94, p < 10^(-9)), while improving geometry via PCA maintains accuracy across architectures (-0.03pp at 95% variance). This relationship is noise-type agnostic -- Gaussian, Uniform, Dropout, and Salt-and-pepper noise all show |r| > 0.90. These results establish that effective dimension provides domain-agnostic predictive and causal information about neural network performance, computed entirely without labels.

Selecting a suitable training dataset is crucial for both general-domain (e.g., GPT-3) and domain-specific (e.g., Codex) language models (LMs). We formalize this data selection problem as selecting a subset of a large raw unlabeled dataset to match a desired target distribution, given some unlabeled target samples. Due to the large scale and dimensionality of the raw text data, existing methods use simple heuristics to select data that are similar to a high-quality reference corpus (e.g., Wikipedia), or leverage experts to manually curate data. Instead, we extend the classic importance resampling approach used in low-dimensions for LM data selection. Crucially, we work in a reduced feature space to make importance weight estimation tractable over the space of text. To determine an appropriate feature space, we first show that KL reduction, a data metric that measures the proximity between selected data and the target in a feature space, has high correlation with average accuracy on 8 downstream tasks (r=0.89) when computed with simple n-gram features. From this observation, we present Data Selection with Importance Resampling (DSIR), an efficient and scalable algorithm that estimates importance weights in a reduced feature space (e.g., n-gram features in our instantiation) and selects data with importance resampling according to these weights. When training general-domain models (target is Wikipedia + books), DSIR improves over random selection and heuristic filtering baselines by 2--2.5% on the GLUE benchmark. When performing continued pretraining towards a specific domain, DSIR performs comparably to expert curated data across 8 target distributions.

The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of n points in d-dimensional Euclidean space into optimal k=O(varepsilon^{-2} ln n) dimensions, while preserving all pairwise distances to within a factor (1 pm varepsilon). The Fast JL transform supports computing the embedding of a data point in O(d ln d +k ln^2 n) time, where the d ln d term comes from multiplication with a d times d Hadamard matrix and the k ln^2 n term comes from multiplication with a sparse k times d matrix. Despite the Fast JL transform being more than a decade old, it is one of the fastest dimensionality reduction techniques for many tradeoffs between varepsilon, d and n. In this work, we give a surprising new analysis of the Fast JL transform, showing that the k ln^2 n term in the embedding time can be improved to (k ln^2 n)/alpha for an alpha = Omega(min{varepsilon^{-1}ln(1/varepsilon), ln n}). The improvement follows by using an even sparser matrix. We also complement our improved analysis with a lower bound showing that our new analysis is in fact tight.

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate dimension reduction, typically achieved via basis expansions. Currently, these bases are chosen a priori without the information for the task at hand and thus may not be effective for the designated task. We instead propose to adaptively learn these bases in an end-to-end fashion. We introduce neural networks that employ a new Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. Our architecture learns to apply parsimonious dimension reduction to functional inputs that focuses only on information relevant to the target rather than irrelevant variation in the input function. Across numerous classification/regression tasks with functional data, our method empirically outperforms other types of neural networks, and we prove that our approach is statistically consistent with low generalization error. Code is available at: https://github.com/jwyyy/AdaFNN.

A ubiquitous feature of data of our era is their extra-large sizes and dimensions. Analyzing such high-dimensional data poses significant challenges, since the feature dimension is often much larger than the sample size. This thesis introduces robust and computationally efficient methods to address several common challenges associated with high-dimensional data. In my first manuscript, I propose a coherent approach to variable screening that accommodates nonlinear associations. I develop a novel variable screening method that transcends traditional linear assumptions by leveraging mutual information, with an intended application in neuroimaging data. This approach allows for accurate identification of important variables by capturing nonlinear as well as linear relationships between the outcome and covariates. Building on this foundation, I develop new optimization methods for sparse estimation using nonconvex penalties in my second manuscript. These methods address notable challenges in current statistical computing practices, facilitating computationally efficient and robust analyses of complex datasets. The proposed method can be applied to a general class of optimization problems. In my third manuscript, I contribute to robust modeling of high-dimensional correlated observations by developing a mixed-effects model based on Tsallis power-law entropy maximization and discussed the theoretical properties of such distribution. This model surpasses the constraints of conventional Gaussian models by accommodating a broader class of distributions with enhanced robustness to outliers. Additionally, I develop a proximal nonlinear conjugate gradient algorithm that accelerates convergence while maintaining numerical stability, along with rigorous statistical properties for the proposed framework.

Dimensionality reduction is critical for deploying dense retrieval systems at scale, yet mainstream post-hoc methods face a fundamental trade-off: principal component analysis (PCA) preserves dominant variance but underutilizes representational capacity, while whitening enforces isotropy at the cost of amplifying noise in the heavy-tailed eigenspectrum of retrieval embeddings. Intermediate spectral scaling methods unify these extremes by reweighting dimensions with a power coefficient γ, but treat γ as a fixed hyperparameter that requires task-specific tuning. We show that the optimal scaling strength γ is not a global constant: it varies systematically with target dimensionality k and is governed by the signal-to-noise ratio (SNR) of the retained subspace. Based on this insight, we propose Spectral Tempering (SpecTemp), a learning-free method that derives an adaptive γ(k) directly from the corpus eigenspectrum using local SNR analysis and knee-point normalization, requiring no labeled data or validation-based search. Extensive experiments demonstrate that Spectral Tempering consistently achieves near-oracle performance relative to grid-searched γ^*(k) while remaining fully learning-free and model-agnostic. Our code is publicly available at https://anonymous.4open.science/r/SpecTemp-0D37.

Recent work has proposed the linear representation hypothesis: that language models perform computation by manipulating one-dimensional representations of concepts ("features") in activation space. In contrast, we explore whether some language model representations may be inherently multi-dimensional. We begin by developing a rigorous definition of irreducible multi-dimensional features based on whether they can be decomposed into either independent or non-co-occurring lower-dimensional features. Motivated by these definitions, we design a scalable method that uses sparse autoencoders to automatically find multi-dimensional features in GPT-2 and Mistral 7B. These auto-discovered features include strikingly interpretable examples, e.g. circular features representing days of the week and months of the year. We identify tasks where these exact circles are used to solve computational problems involving modular arithmetic in days of the week and months of the year. Finally, we provide evidence that these circular features are indeed the fundamental unit of computation in these tasks with intervention experiments on Mistral 7B and Llama 3 8B, and we find further circular representations by breaking down the hidden states for these tasks into interpretable components.

Recent advances in diffusion and flow matching models have highlighted a shift in the preferred prediction target -- moving from noise (varepsilon) and velocity (v) to direct data (x) prediction -- particularly in high-dimensional settings. However, a formal explanation of why the optimal target depends on the specific properties of the data remains elusive. In this work, we provide a theoretical framework based on a generalized prediction formulation that accommodates arbitrary output targets, of which varepsilon-, v-, and x-prediction are special cases. We derive the analytical relationship between data's geometry and the optimal prediction target, offering a rigorous justification for why x-prediction becomes superior when the ambient dimension significantly exceeds the data's intrinsic dimension. Furthermore, while our theory identifies dimensionality as the governing factor for the optimal prediction target, the intrinsic dimension of manifold-bound data is typically intractable to estimate in practice. To bridge this gap, we propose k-Diff, a framework that employs a data-driven approach to learn the optimal prediction parameter k directly from data, bypassing the need for explicit dimension estimation. Extensive experiments in both latent-space and pixel-space image generation demonstrate that k-Diff consistently outperforms fixed-target baselines across varying architectures and data scales, providing a principled and automated approach to enhancing generative performance.

A crucial component of many deep learning applications (such as FAQ and RAG) is dense retrieval, in which embedding models are used to convert raw text to numerical vectors and then get the most similar text by MIPS (Maximum Inner Product Search). Some text embedding benchmarks (e.g. MTEB, BEIR, and AIR-Bench) have been established to evaluate embedding models accurately. Thanks to these benchmarks, we can use SOTA models; however, the deployment and application of these models in industry were hampered by their large vector dimensions and numerous parameters. To alleviate this problem, 1) we present a distillation technique that can enable a smaller student model to achieve good performance. 2) Inspired by MRL we present a training approach of reducing the vector dimensions based on its own vectors or its teacher vectors. 3) We do simple yet effective alignment training between images and text to make our model a multimodal encoder. We trained Stella and Jasper models using the technologies above and achieved high scores on the MTEB leaderboard. We release the model and data at Hugging Face Hub (https://huggingface.co/infgrad/jasper_en_vision_language_v1) and the training logs are at https://api.wandb.ai/links/dunnzhang0/z8jqoqpb.

Diffusion models have emerged as the new state-of-the-art generative model with high quality samples, with intriguing properties such as mode coverage and high flexibility. They have also been shown to be effective inverse problem solvers, acting as the prior of the distribution, while the information of the forward model can be granted at the sampling stage. Nonetheless, as the generative process remains in the same high dimensional (i.e. identical to data dimension) space, the models have not been extended to 3D inverse problems due to the extremely high memory and computational cost. In this paper, we combine the ideas from the conventional model-based iterative reconstruction with the modern diffusion models, which leads to a highly effective method for solving 3D medical image reconstruction tasks such as sparse-view tomography, limited angle tomography, compressed sensing MRI from pre-trained 2D diffusion models. In essence, we propose to augment the 2D diffusion prior with a model-based prior in the remaining direction at test time, such that one can achieve coherent reconstructions across all dimensions. Our method can be run in a single commodity GPU, and establishes the new state-of-the-art, showing that the proposed method can perform reconstructions of high fidelity and accuracy even in the most extreme cases (e.g. 2-view 3D tomography). We further reveal that the generalization capacity of the proposed method is surprisingly high, and can be used to reconstruct volumes that are entirely different from the training dataset.

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

Traditional retrieval methods have been essential for assessing document similarity but struggle with capturing semantic nuances. Despite advancements in latent semantic analysis (LSA) and deep learning, achieving comprehensive semantic understanding and accurate retrieval remains challenging due to high dimensionality and semantic gaps. The above challenges call for new techniques to effectively reduce the dimensions and close the semantic gaps. To this end, we propose VectorSearch, which leverages advanced algorithms, embeddings, and indexing techniques for refined retrieval. By utilizing innovative multi-vector search operations and encoding searches with advanced language models, our approach significantly improves retrieval accuracy. Experiments on real-world datasets show that VectorSearch outperforms baseline metrics, demonstrating its efficacy for large-scale retrieval tasks.

The autoencoder is an unsupervised learning paradigm that aims to create a compact latent representation of data by minimizing the reconstruction loss. However, it tends to overlook the fact that most data (images) are embedded in a lower-dimensional space, which is crucial for effective data representation. To address this limitation, we propose a novel approach called Low-Rank Autoencoder (LoRAE). In LoRAE, we incorporated a low-rank regularizer to adaptively reconstruct a low-dimensional latent space while preserving the basic objective of an autoencoder. This helps embed the data in a lower-dimensional space while preserving important information. It is a simple autoencoder extension that learns low-rank latent space. Theoretically, we establish a tighter error bound for our model. Empirically, our model's superiority shines through various tasks such as image generation and downstream classification. Both theoretical and practical outcomes highlight the importance of acquiring low-dimensional embeddings.

The analysis of emotions expressed in text has numerous applications. In contrast to categorical analysis, focused on classifying emotions according to a pre-defined set of common classes, dimensional approaches can offer a more nuanced way to distinguish between different emotions. Still, dimensional methods have been less studied in the literature. Considering a valence-arousal dimensional space, this work assesses the use of pre-trained Transformers to predict these two dimensions on a continuous scale, with input texts from multiple languages and domains. We specifically combined multiple annotated datasets from previous studies, corresponding to either emotional lexica or short text documents, and evaluated models of multiple sizes and trained under different settings. Our results show that model size can have a significant impact on the quality of predictions, and that by fine-tuning a large model we can confidently predict valence and arousal in multiple languages. We make available the code, models, and supporting data.

We propose a game-based formulation for learning dimensionality-reducing representations of feature vectors, when only a prior knowledge on future prediction tasks is available. In this game, the first player chooses a representation, and then the second player adversarially chooses a prediction task from a given class, representing the prior knowledge. The first player aims is to minimize, and the second player to maximize, the regret: The minimal prediction loss using the representation, compared to the same loss using the original features. For the canonical setting in which the representation, the response to predict and the predictors are all linear functions, and under the mean squared error loss function, we derive the theoretically optimal representation in pure strategies, which shows the effectiveness of the prior knowledge, and the optimal regret in mixed strategies, which shows the usefulness of randomizing the representation. For general representations and loss functions, we propose an efficient algorithm to optimize a randomized representation. The algorithm only requires the gradients of the loss function, and is based on incrementally adding a representation rule to a mixture of such rules.

This paper discusses about a sorting algorithm which uses the concept of buckets where each bucket represents a certain number of digits. A two dimensional data structure is used where one dimension represents buckets i. e; number of digits and each bucket's corresponding dimensions represents the input numbers that belong to that bucket. Each bucket is then individually sorted. Since every preceding bucket elements will always be smaller than the succeeding buckets no comparison between them is required. By doing this we can significantly reduced the time complexity of any sorting algorithm used to sort the given set of inputs.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Making statements about the performance of trained models on tasks involving new data is one of the primary goals of machine learning, i.e., to understand the generalization power of a model. Various capacity measures try to capture this ability, but usually fall short in explaining important characteristics of models that we observe in practice. In this study, we propose the local effective dimension as a capacity measure which seems to correlate well with generalization error on standard data sets. Importantly, we prove that the local effective dimension bounds the generalization error and discuss the aptness of this capacity measure for machine learning models.