The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.

Data augmentation is a simple and model-agnostic approach for exploiting known invariances in learning problems. Given a group acting on the input space, one augments the training set with transformed copies of each sample. Because it exploits symmetries without modifying the underlying learning algorithm, data augmentation can be applied broadly across learning methods. However, this universality comes at a computational cost: when the group is large, full group-sized augmentation quickly becomes computationally infeasible. This raises a fundamental question: Can partial data augmentation achieve the same statistical benefits as full augmentation in terms of generalization and sample complexity? We develop a general framework for investigating this question using Fourier analysis and the representation theory of finite groups. We show that, for a broad class of classical learning problems, partial data augmentation based on a randomly sampled subset of group elements achieves the same minimax rates as full augmentation, up to an approximation error that vanishes as the subset size increases. Our results provide a theoretical explanation for why partial augmentation can retain the statistical benefits of full augmentation despite enforcing symmetry only approximately, and shed light on a recently raised question in learning with symmetries: whether statistically optimal learning under general group invariances can be achieved using computationally scalable methods. Moreover, we prove a complementary impossibility result: enforcing exact invariance via data augmentation requires averaging over the entire group, and cannot be achieved by any strict subset when the hypothesis space is sufficiently expressive. Together, these results provide a unified perspective on full and partial data augmentation, as well as exact and approximate symmetry enforcement.

In this paper, we address the challenges associated with merging low-rank adaptations of large neural networks. With the rise of parameter-efficient adaptation techniques, such as Low-Rank Adaptation (LoRA), model fine-tuning has become more accessible. While fine-tuning models with LoRA is highly efficient, existing merging methods often sacrifice this efficiency by merging fully-sized weight matrices. We propose the Core Space merging framework, which enables the merging of LoRA-adapted models within a common alignment basis, thereby preserving the efficiency of low-rank adaptation while substantially improving accuracy across tasks. We further provide a formal proof that projection into Core Space ensures no loss of information and provide a complexity analysis showing the efficiency gains. Extensive empirical results demonstrate that Core Space significantly improves existing merging techniques and achieves state-of-the-art results on both vision and language tasks while utilizing a fraction of the computational resources.

Vector-Quantized Variational Autoencoders (VQVAEs) have enabled strong performance in generative modeling by mapping continuous data to learnable codes. In this work, we identify a surprising yet consistent phenomenon that we term dimensional collapse: despite using high-dimensional embeddings, VQVAEs tend to compress their representations into a much smaller subspace, typically only 4 to 10 dimensions. We provide an in-depth analysis of this phenomenon and reveal its relation to model performance and learning dynamics. Interestingly, VQVAEs naturally gravitate toward this low-dimensional regime, and enforcing higher-dimensional usage (e.g., via rank regularization) could lead to degraded performance. To overcome this low-dimensionality limitation, we propose Divide-and-Conquer VQ (DCVQ), which partitions the latent space into multiple low-dimensional subspaces, each quantized independently. By design, each subspace respects the model's preference for low dimensionality, while their combination expands the overall capacity. Our results show that DCVQ overcomes the inherent dimensional bottleneck and achieves improved reconstruction quality across image datasets.

Transformer-based language models exhibit complex and distributed behavior, yet their internal computations remain poorly understood. Existing mechanistic interpretability methods typically treat attention heads and multilayer perceptron layers (MLPs) (the building blocks of a transformer architecture) as indivisible units, overlooking possibilities of functional substructure learned within them. In this work, we introduce a more fine-grained perspective that decomposes these components into orthogonal singular directions, revealing superposed and independent computations within a single head or MLP. We validate our perspective on widely used standard tasks like Indirect Object Identification (IOI), Gender Pronoun (GP), and Greater Than (GT), showing that previously identified canonical functional heads, such as the "name mover," encode multiple overlapping subfunctions aligned with distinct singular directions. Nodes in a computational graph, that are previously identified as circuit elements show strong activation along specific low-rank directions, suggesting that meaningful computations reside in compact subspaces. While some directions remain challenging to interpret fully, our results highlight that transformer computations are more distributed, structured, and compositional than previously assumed. This perspective opens new avenues for fine-grained mechanistic interpretability and a deeper understanding of model internals.

Attention mechanisms have achieved significant empirical success in multiple fields, but their underlying optimization objectives remain unclear yet. Moreover, the quadratic complexity of self-attention has become increasingly prohibitive. Although interpretability and efficiency are two mutually reinforcing pursuits, prior work typically investigates them separately. In this paper, we propose a unified optimization objective that derives inherently interpretable and efficient attention mechanisms through algorithm unrolling. Precisely, we construct a gradient step of the proposed objective with a set of forward-pass operations of our Contractand-Broadcast Self-Attention (CBSA), which compresses input tokens towards low-dimensional structures by contracting a few representatives of them. This novel mechanism can not only scale linearly by fixing the number of representatives, but also covers the instantiations of varied attention mechanisms when using different sets of representatives. We conduct extensive experiments to demonstrate comparable performance and superior advantages over black-box attention mechanisms on visual tasks.

LLMs have demonstrated unprecedented capabilities in natural language processing, yet their practical deployment remains hindered by persistent factuality and faithfulness hallucinations. While existing methods address these hallucination types independently, they inadvertently induce performance trade-offs, as interventions targeting one type often exacerbate the other. Through empirical and theoretical analysis of activation space dynamics in LLMs, we reveal that these hallucination categories share overlapping subspaces within neural representations, presenting an opportunity for concurrent mitigation. To harness this insight, we propose SPACE, a unified framework that jointly enhances factuality and faithfulness by editing shared activation subspaces. SPACE establishes a geometric foundation for shared subspace existence through dual-task feature modeling, then identifies and edits these subspaces via a hybrid probe strategy combining spectral clustering and attention head saliency scoring. Experimental results across multiple benchmark datasets demonstrate the superiority of our approach.

Generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and diffusion models often suffer from mode collapse, failing to reproduce the full diversity of their training data. While this problem has been extensively studied in image generation, it remains largely unaddressed for time series. We introduce a formal definition of mode collapse for time series and propose DMD-GEN, a geometry-aware metric that quantifies its severity. DMD-GEN leverages Dynamic Mode Decomposition (DMD) to extract coherent temporal structures and uses Optimal Transport between DMD eigenvectors to measure discrepancies in underlying dynamics. By representing the subspaces spanned by the DMD eigenvectors as point structures on a Grassmann manifold, and comparing them via Wasserstein distances computed from principal angles, DMD-GEN enables a principled geometric comparison between real and generated sequences. The metric is efficient, requiring no additional training, supports minibatch evaluation, and is easily parallelizable. Beyond quantification, DMD-GEN offers interpretability by revealing which dynamical modes are distorted or missing in the generated data.

Vision-Language Models (VLMs) excel at zero-shot inference but often degrade under test-time domain shifts. For this reason, episodic test-time adaptation strategies have recently emerged as powerful techniques for adapting VLMs to a single unlabeled image. However, existing adaptation strategies, such as test-time prompt tuning, typically require backpropagating through large encoder weights or altering core model components. In this work, we introduce Spectrum-Aware Test-Time Steering (STS), a lightweight adaptation framework that extracts a spectral subspace from the textual embeddings to define principal semantic directions and learns to steer latent representations in a spectrum-aware manner by adapting a small number of per-sample shift parameters to minimize entropy across augmented views. STS operates entirely at inference in the latent space, without backpropagation through or modification of the frozen encoders. Building on standard evaluation protocols, our comprehensive experiments demonstrate that STS largely surpasses or compares favorably against state-of-the-art test-time adaptation methods, while introducing only a handful of additional parameters and achieving inference speeds up to 8 faster with a 12 smaller memory footprint than conventional test-time prompt tuning. The code is available at https://github.com/kdafnis/STS.

Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm (ADASAP) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. ADASAPoutperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, ADASAPscales to a dataset with > 3 108 samples, a feat which has not been accomplished in the literature.