Neural networks are known to be susceptible to over-reliance on spurious correlations. However, the precise mechanism by which models exploit shortcut features is not fully understood, and algorithms to mitigate this behavior rely on as yet unjustified assumptions about the learned representations. In this work, we provide the first end-to-end theoretical characterization of spurious feature learning for two-layer ReLU neural networks trained by online minibatch SGD on the logistic loss. We consider data drawn from the high-dimensional Boolean hypercube with a quadratic signal function (namely XOR) and a linear spurious correlation. We show that SGD learns the spurious feature first, and exponentially fast. Moreover, the optimization dynamics couple the spurious and signal features, with a stronger spurious component inhibiting signal feature learning. Our analysis reveals precise phase transitions in the learning dynamics. In the first phase, alignment between the signs of the spurious feature and second-layer weight drives rapid growth of the spurious feature. In the second phase, large majority group margin slows learning and the signal feature remains suppressed. When the spurious correlation is maximally strong, we show theoretically that the spurious feature dominates even at the sample complexity threshold where XOR would be learned in isolation (i.e., if the spurious feature was absent). In contrast, when the correlation strength is constant, we provide preliminary empirical evidence that the model can eventually learn the XOR signal, although the spurious feature is not forgotten.

Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single $τ\in [0, 1]$ that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that $τ$ provides.

In recent years, models based on the Transformer architecture have seen widespread applications and have become one of the core tools in the field of deep learning. Numerous successful techniques, such as parameter-efficient fine-tuning and efficient scaling, have been proposed surrounding their applications to further enhance performance. However, the success of these strategies has always lacked the support of rigorous mathematical theory. To study the underlying mechanisms behind Transformers and related techniques, we first propose a Transformer learning framework motivated by distribution regression, with distributions being inputs, connect a two-stage sampling process with natural language processing, and present a mathematical formulation of the attention mechanism called attention operator. We demonstrate that by the attention operator, Transformers can compress distributions into function representations without loss of information. Moreover, with the advantages of our novel attention operator, Transformers exhibit a stronger capability to learn functionals with more complex structures than convolutional neural networks and fully connected networks. Finally, we obtain a generalization bound within the distribution regression framework. Through the aforementioned theoretical results, we further discuss some successful techniques emerging with large language models (LLMs), such as prompt tuning, parameter-efficient fine-tuning, and efficient scaling. We also provide theoretical insights behind these techniques within our novel analysis framework.

Data valuation quantifies the intrinsic quality of individual samples to enable principled data curation, quality control, and robust learning. For time series in critical domains such as healthcare, finance, and industrial monitoring, effective valuation methods are essential yet fundamentally lacking. Existing approaches are either model-dependent, limiting their generalizability, or designed for i.i.d. data and thus fail to capture temporal dependencies, multi-scale patterns, and non-stationary dynamics inherent to sequential data. We introduce TimeLAVA, a learning-agnostic framework that values temporal segments by their marginal contribution to minimizing distributional discrepancy between evaluated and reference data. At its core is a novel Selective Wavelet-based Wasserstein discrepancy combining multi-scale wavelet transforms for temporal localization with unbalanced optimal transport for robustness to distributional shifts. Segment values are efficiently computed via sensitivity analysis without requiring model training and aggregated into point-wise scores. We provide theoretical guarantees linking valuation to model-agnostic generalization and prove bounded sensitivity to outlier contamination. Extensive experiments across anomaly detection, data pruning, and label noise detection demonstrate that TimeLAVA produces significantly more informative value scores than existing methods on diverse real-world datasets.

Effective sub-typing (also known as grouping or clustering) of patients using their electronic health record (EHR) data can greatly inform precision medicine efforts. However, subtyping temporal EHR datasets is known to be challenging due to inherent EHR issues, including complexity and irregularity. In this study, we propose a self-supervised Mamba-based model that learns effective EHR representations and enables enhanced patient subtyping. We evaluate the proposed model on public and private real-world EHR datasets to classify the data based on the available labels and subtype patients based on the representations learned from the model. Through an extensive set of experiments, we demonstrate that our model's design choices lead to better performance compared to competitive baseline models for prediction. Moreover, we evaluate several clustering techniques to demonstrate that our findings offer valuable insights into subtyping patients based on temporal records from EHR models\footnote{Our implementations are available at https://github.com/healthylaife/triplet_mamba.

The common factor analytic model is related to Helmholtz and Boltzmann machines, can be conceived as a linear autoencoder, or can be thought of as a single-hidden-layer generative neural network. We thus consider it a basal generative representation learner that can be used as a minimal model for studying the foundational characteristics of (deep) generative model architectures. We focus on the fundamental problem of indeterminacy in latent factor projections. This indeterminacy implies that, even when the intrinsic dimension of the latent vector is known, regularity conditions are met, and rotational indeterminacy is resolved, an inherent indefiniteness in the retrieval of causative latent sources remains: they will be uncertain, distributionally deviant, and non-unique. This can have major implications for data representation but remains an elusive issue, even to practitioners and theorists well-versed in the factor model. Moreover, this classic psychometric problem is intricately related to the modern issue of latent variable collapse in the variational autoencoder framework for deep generative modeling. Here, we assess this indeterminacy from various perspectives and show how these are mathematically and conceptually related and we discuss subsequent implications for the Psychometrics, Statistics, and Artificial Intelligence communities. We show that one has latent factor determinacy across all its facets when the feature-dimension grows to infinity. This feeds into an essentially distribution-free estimation approach in the sample case when the number of features grows very large. We conclude, as these are emergent properties at scale, that the factor model is suited for representation learning of very-high-dimensional data.

Large language models (LLMs) are increasingly deployed in high-stakes domains, where free-text explanations such as chain-of-thought and post-hoc rationales are used to justify model outputs. Yet it remains unclear whether these explanations are sufficient, i.e., if they contain enough information to explain the model's output-generating process. We generalize classical sufficiency from feature attributions to arbitrary explanations and prove that explanation sufficiency can change depending on the input distribution, which must be explicitly defined for LLM explanations. We propose using the LLM itself to generate alternative inputs conditioned on an explanation, capturing its beliefs about possible inputs. We formalize self-consistent sufficiency as a goal for free-text explanations and introduce an information-theoretic metric, SCSuff, that enables evaluation of free-text explanations without relying on predefined biases or shortcuts. Our experiments show that SCSuff agrees with targeted perturbation tests where applicable and demonstrate that explanation sufficiency can vary with the input distribution. We find LLM explanations are generally insufficient and weakly correlated with model size, accuracy, or output entropy. Analysis of final-token hidden states shows that top and bottom SCSuff scores can be predicted from internal representations, suggesting that SCSuff can guide detection and improvement of sufficient LLM explanations. The code for this paper is available at https://github.com/rajesh-lab/self-consistent-sufficiency .

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., $\{+,\times,\sin,\exp\}$ and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth $d$ and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size $K=|\mathcal{H}_{\mathrm{base}}|$ together with Maurer-type vector contraction yields $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{d}) \leq (Kb\sqrt{2}L)^{d-1}\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})$ with arity bound $b$; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $K,b=O(1)$ and $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})=O(n^{-1/2})$. We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term $(\widehat{L}^{d})/\sqrt{n}$.

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.