Neural networks trained with gradient-based methods exhibit a strong simplicity bias: they learn simpler statistical features of their data before moving to more complex features. Previous analyses of this phenomenon have largely focused on settings with (quasi-)isotropic inputs. In this work, we study the simplicity bias from a Fourier perspective, which allows us to include two key features of natural images in the analysis: approximate translation-invariance and power-law spectra. We first show experimentally that simple neural networks trained on image classification tasks first rely on amplitude information -- related to pair-wise correlations between pixels -- before exploiting phase information, which encodes edges and higher-order correlations. In view of this, we introduce a synthetic data model for translation-invariant inputs that allows precise control over amplitudes and phases while remaining tractable. We rigorously establish that for isotropic and high-dimensional inputs, classification based on phase information alone is a genuinely hard task: online stochastic gradient descent (SGD) cannot distinguish the structured inputs from noise within $n \ll N^3$ steps, but needs at least $n \gg N^3 \log^2{N}$ steps. In contrast, we show both experimentally and theoretically that power-law spectra can dramatically accelerate the speed of learning phase information, even if the spectra do not help with classification. Simulations with two-layer networks trained on textures and with deep convolutional networks on ImageNet and CIFAR100 confirm this non-trivial interaction between amplitudes and phases, providing mechanistic insights into how deep neural networks can learn natural image distributions efficiently.

Many decision-facing stochastic systems are observed through aggregate distributions rather than scalar trajectories: queue occupancies, mobility shares, publichealth mixtures, generation-source shares, ecological compositions, and air-quality severity profiles all live on the probability simplex and evolve over time. We study causal (time-respecting online) forecasting for these distribution-valued time series and argue that the transition operator itself should be structured around the simplex. We introduce CAST (Causal Anchored Simplex Transport), a successor-local operator that (i) retrieves empirical successors from causal context, (ii) stabilizes them with a persistence anchor, and (iii) applies a bounded local stochastic transport on ordered supports; every stage preserves the simplex by construction. We identify a structural failure mode, latent transition-kernel aliasing, where similar observed distributions evolve differently under different contextual regimes, and prove that any forecaster depending only on an aliased summary incurs an irreducible weighted Jensen-Shannon excess-risk lower bound, while the CAST hypothesis class contains the regime-aware Bayes successor; for ordered supports an additional Pinsker separation holds whenever the transported successor lies outside the no-transport anchor hull. On a suite of eleven public and simulated benchmarks spanning ecology, energy, diet, mortality, employment, air quality, severe weather, mobility, and G/G/1, Gt/G/1 queue occupancy, CAST achieves the best average rank on both one-step KL (1.27) and autoregressive rollout JSD (1.91), winning 8/11 sections on each metric against a broad statistical, compositional, recurrent, convolutional, Transformer, and modern time-series baseline set, and top-2 on all 11 sections for offline KL. Component ablations and a controlled synthetic aliasing experiment corroborate the theory. The code release is available at this link.

However, many real-world problems involve intrinsic uncertainties arising from incomplete physical knowledge, imperfect observations, environmental variability, and unresolved multiscale processes. These uncertainties may appear, for example, in initial or boundary conditions, unresolved physical processes, or heterogeneous material properties, and can significantly impact predictive accuracy. To obtain reliable and uncertainty-aware predictions, such effects are often incorporated directly into the mathematical formulation through random inputs, stochastic coefficients, or stochastic perturbations, leading to stochastic partial differential equations (SPDEs). Deriving numerical solutions to SPDEs has thus become a central focus of the uncertainty quantification (UQ) community, where extensive efforts have been devoted to developing efficient solvers that can accurately characterize and propagate uncertainty in high-dimensional, nonlinear dynamical systems (see, e.g., [1, 2, 12, 9, 13, 14, 18, 30, 40, 50, 56] and the references therein). Although traditional methods are effective for solving SPDEs and propagating uncertainty from stochastic input data to model predictions, they often require substantial computational cost, especially for time-dependent and multiscale problems [49, 51]. 1

We study the multi-task linear regression problem in the presence of contaminated tasks. We address the setting where the unknown parameters of a majority of tasks are close in the $\ell_2$-norm, while a fraction of tasks are arbitrary outliers. Existing theoretical frameworks for this problem rely heavily on the assumption that the empirical second moment of each task has a minimum eigenvalue bounded away from zero (order $Ω(1)$). Crucially, this assumption fails in many high-dimensional scenarios, rendering prior guarantees vacuous. To overcome this limitation, we propose an estimator based on matrix-weighted norm regularization. We also introduce a relative balancedness condition, quantified by a balancedness constant, that compares each task's second moment with the average inlier geometry and relaxes the need for taskwise second-moment lower bounds. In favorable regimes with moderate balancedness, our prediction MSE bounds match the rate of Duan and Wang (2023) under substantially weaker spectral assumptions; the resulting task-overall MSE is minimax optimal up to logarithmic factors. Furthermore, we demonstrate that our estimator enjoys a safety guarantee: when the relevant balancedness constant is large or infinite, or when tasks are unrelated, the method performs no worse than independent task learning. Consequently, our methodology achieves simultaneous adaptivity to task similarity, robustness to outliers, and safety outside favorable transfer regimes.

Many practical data analysis tasks reduce to learning, from observed samples, how a collection of variables depend on each other. A widely used approach is to fit a Gaussian graphical model, which represents the dependence structure as a graph connecting the variables. In a number of important applications, such as financial returns, gene co-expression, and climate or network analysis, the dependencies tend to be positive: variables move together rather than offset each other. Encoding this positivity through the constraint of multivariate total positivity of order two (MTP2) yields an attractive estimator that produces accurate fits with no tuning required. The resulting graphs are, however, typically much denser than the underlying ground-truth model, which makes them hard to interpret and slow to use in any downstream task that operates on the graph. In this work, we propose a novel highly-scalable approach for learning Gaussian graphical models from data using spectral sparsification; we call it Spectral-MTP2. Spectral graph sparsification is a fundamental method which aims to preserve meaningful properties of a dense graph with a sparser subgraph. We theoretically and empirically investigate and validate our method, and show that learning Gaussian Graphical Models under MTP2 using spectral sparsification preserves MTP2 and approximates well the original model in terms of Kullback-Leibler divergence and Gaussian log-likelihood. In simulations and applications to equity returns and gene expression, we observe that Spectral-MTP2 retains most of the fit quality of the denser MTP2 baseline, while producing substantially sparser and more interpretable graphs.

Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and generalization properties of neural models. In this work, we show that in the high-dimensional regime, stochastic gradient descent on diagonal linear networks is well-approximated by continuous dynamics governed by a stochastic differential equation (SDE), which explicitly decouples the drift from the gradient noise. We further derive a deterministic partial differential equation whose solution propagates the relevant state of the iterates and characterizes the time evolution of a broad class of observable statistics, including the risk, curvature, and other metrics for optimality. Finally, we show that, under a suitable parametrization, the stochastic dynamics are globally well posed and converge exponentially fast to zero risk with high probability, yielding a fully explicit non-asymptotic description of their long-time behavior. Numerical simulations corroborate our theoretical findings.

We develop a geometric theory of projection heads in self-supervised learning by modeling the head as a trainable Riemannian metric on the backbone representation manifold. We show that linear heads perform implicit subspace whitening, while nonlinear heads adapt local metrics to satisfy the specific topological constraints of the loss, with head depth empirically dictating this capacity. Analyzing dimensional collapse, we prove that smooth nonlinear heads natively induce negative eigenvalues in the Hessian at collapsed equilibria, making them unstable. We empirically validate this by continuously tracking the optimization geometry during training, which reveals that smooth activations like Swish can generate explicit negative curvature to escape collapse, whereas linear and ReLU heads under continuous-time gradient flow cannot, relying instead on discrete-time optimization dynamics and BatchNorm. Finally, we geometrically characterize how metric degeneracy governs the information-invariance trade-off, explaining why the head must be discarded. Evaluated across contrastive and decorrelation-based objectives on foundation models, our results demonstrate that the projection head acts as a universal geometric buffer, decoupling the semantic backbone from the rigid, destructive constraints of the pretraining objective.

Low-rank matrix completion is a widely studied problem with many variants. Inductive matrix completion (IMC) incorporates row and column side information to significantly narrow the search space. Prior work falls into two regimes: methods that exploit this structure to achieve reduced sample complexity but only in noiseless settings, and methods that handle noise but require sample complexity matching the ambient matrix dimension, forfeiting the sample efficiency that side information should provide. In this paper, we close this gap by studying noisy IMC with a nonconvex projected gradient descent algorithm with spectral initialization. Our main technical contribution is establishing a regularity condition for the IMC loss function that holds at the reduced sample complexity determined by the effective problem size, scaling with the side information dimension a rather than the ambient dimension n. This directly yields linear convergence and an estimation error that both depend only on the effective problem size rather than the ambient matrix dimension. We further extend our analysis to the inexact side information setting, demonstrating that the reduced sample complexity is maintained and the estimation error is order-optimal with respect to the inexactness of the side information. Extensive simulations and real-world experiments on the MovieLens dataset validate our theoretical findings.

Spectral deconvolution is essential for extracting peak structures that encode material properties and chemical structures, but conventional automated methods often fail when spectra contain high-intensity noise or unknown background components. In practice, scientists rarely interpret spectra in isolation. Instead, they identify physically meaningful peaks by relating spectral structures to auxiliary information such as physical-property values, chemical structures, and trends across related measurements. Here, we propose a Bayesian framework that integrates spectral deconvolution with a model of expert scientific reasoning. In this work, expert scientific reasoning refers to the practice of evaluating candidate spectral structures by their consistency with independently measured physical-property values, rather than to manual expert intervention during inference. We formalize this reasoning as a physical-property regression layer, implemented using Gaussian process regression, and couple it with Bayesian spectral deconvolution. By averaging the physical-property likelihood over posterior predictive spectra inferred from Bayesian spectral deconvolution, the proposed method selects spectral models according to the consistency between inferred spectral structures and physical-property information. We validate the framework using synthetic spectra with high-intensity noise or unknown backgrounds and infrared spectra of poly(lactic acid). The method recovers physically meaningful peak structures that conventional Bayesian spectral deconvolution misses or misidentifies from spectra alone, including weak peaks in poly(lactic acid) IR spectra related to measured degradation rates. These results demonstrate that integrating expert scientific reasoning with Bayesian spectral deconvolution enables robust peak estimation under conditions where spectrum-only inference is unreliable.

Large-scale hypothesis testing is central to modern science, where controlling the False Discovery Rate (FDR) has become the standard approach to managing false positives across many simultaneous tests. Hypotheses rarely exist in isolation; they often exhibit structure through proximity, connectivity, or hierarchy. This structure represents both a challenge and an opportunity: while classical methods treat these dependencies as obstacles requiring conservative correction, leveraging them can substantially increase discovery power. Here, we reframe structured FDR control as a regularized learning problem. By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR. We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.