Tensor-valued data arise naturally in neuroimaging, genomics, climate science, and spatiotemporal networks, where multilinear dependencies across modes carry information that is destroyed under vectorization. Existing approaches either impose a single low-rank structure, which can miss localized signal, or treat the tensor as a long vector, which discards its multiway geometry. We propose a *Dual-Channel Tensor Neural Network* (DC-TNN) that decomposes each tensor input into a low-rank core and a sparse refinement, and processes the two components through coupled neural channels. The framework is structure-agnostic and accommodates CP, Tucker, and tensor-train cores within a single architecture. For estimation, we establish non-asymptotic risk bounds for the DC-TNN estimator that decompose into network approximation, core estimation, and refinement-selection terms, and show that the effective dimension is determined jointly by the core rank and refinement sparsity rather than by the ambient tensor size. For inference, we develop a *structure-aware conformal ROC* procedure that calibrates within the core-refinement latent space and produces ROC and AUC confidence bands with finite-sample, distribution-free coverage. Building on this, we propose a *conformal structure selector* that, to our knowledge, is the *first distribution-free procedure* for choosing among candidate tensor decompositions with finite-sample validity. Simulations and an analysis of a protein dataset demonstrate competitive predictive accuracy, reliable uncertainty quantification, and consistent recovery of the tensor structure.

The construction of models from data is a significant contributor to the energetic costs of computation. Because of this, understanding how foundational thermodynamic bounds apply to modeling algorithms will be increasingly important. Here, we study the thermodynamic costs of a basic and fundamental modeling algorithm: simple linear regression. Following Landauer, we approximate the thermodynamic lower bound on irreversibly performing both exact linear regression and linear regression via stochastic gradient descent as implemented on floating-point numbers. From this, we derive energycost aware scaling laws for the optimal dataset size for training a linear regression model given a generalization error dependent demand for inference. Additionally, we discuss a method to lower bound the entropy production from the mismatch cost for algorithms with continuous input variables.

Physical activity (PA) plays an important role in maintaining and improving health. Daily steps have been a key PA measure that is easily accessible with common wearable devices. However, methods are lacking to recommend a personalized optimal distribution of daily steps over a period of time for the best of certain health biomarkers. In this paper, we fill this void based on the data from the All of Us Research Program which includes months of step counts as well as repeated measurements of key health biomarkers. We develop a new offline reinforcement learning (RL) algorithm to learn personalized and optimal PA distributions associated with cardiometabolic risk, where the action is a function representing the daily step distribution over a period of time. Simulation studies demonstrate the advantage of the proposed approach over existing continuous-action RL methods. The learned optimal policy from the All of Us data generally suggests people take more daily steps and also follow a more consistent pattern of PA over time while offering tailored recommendations for subgroups in blood glucose level, body mass index, blood pressure, age, and sex.

We introduce DeRegiME -- Deep Regime Mixture of Experts -- a direct multi-horizon probabilistic forecaster that separates latent uncertainty regimes from the underlying signal and softly assigns each forecast location to learned recurring regimes using a sparse variational Gaussian process (GP) whose nonstationary regime-mixing kernel and Student-t likelihood combine per-regime sub-kernels and noise processes via a shared gate. This yields a single sparse-GP posterior, not a mixture of GP experts. DeRegiME addresses a key limitation of neural forecasters: point forecasts discard residual uncertainty, and probabilistic heads -- whether single marginals, uninterpreted mixtures, quantile sets, or diffusion samples -- rarely expose the regime structure of the residual. Yet distribution shift in noisy heteroskedastic time series may be abrupt, gradual, or horizon-dependent and often appears in residual uncertainty rather than the conditional mean. DeRegiME yields an interpretable mean-residual-noise decomposition with a direct-sum feature-space representation that anchors regimes as clusters of residual similarity whose transitions surface as implicit changepoints. The effective number of regimes is pruned by the stick-breaking gate. We prove kernel validity and predictive-density propriety, and across ten benchmarks and three encoder grids DeRegiME improves negative log predictive density (NLPD) by 20.3% over the strongest encoder-matched baseline, a DeepAR/GluonTS-style dynamic Student-t head, with parallel gains on CRPS (3.0%) and MSE (4.7%). Improvements are consistent across all datasets, which span abrupt, gradual, and seasonal shifts.

Sequential prediction is challenging in regimes of delayed disambiguation, where early observations are ambiguous and multiple latent explanations remain plausible until sufficient evidence accumulates. Standard approaches based on marginal inference struggle in this setting, either collapsing uncertainty prematurely or failing to recover once informative evidence arrives. We introduce EviTrack, a test-time inference framework that operates over latent trajectories rather than marginal states. EviTrack maintains a set of competing trajectory hypotheses and applies evidence- and likelihood-ratio-based selection to delay commitment until supported by data, drawing inspiration from hypothesis management in multiple hypothesis tracking and track-before-detect. To evaluate this setting, we construct a controlled synthetic benchmark with known latent ground truth that explicitly exhibits delayed disambiguation. At matched inference budget, EviTrack substantially outperforms sampling-based baselines, achieving faster post-disambiguation recovery. These results show that, in delayed disambiguation regimes, moderate trajectory-level selection is more effective than increasing sampling coverage, highlighting selection over sampling as a key principle for reliable sequential inference.

Stochastic gradient descent (SGD) has been a cornerstone of machine learning since the pioneering work of Robbins & Monro (1951). Beyond its algorithmic simplicity and scalability, SGD has also become a central object of theoretical study, with refined analyses linking its dynamics to implicit regularization, generalization performance, and algorithmic stability. For decades, theoretical analyses of SGD have largely resided within the realm of classical stochastic approximation (Polyak & Juditsky, 1992; Lai, 2003; Bottou et al., 2018), where the data dimension is considered fixed while the sample size tends to infinity. While this regime has yielded foundational insights, it no longer fully reflects the characteristics of modern learning systems. Contemporary applications often operate in regimes where data dimension, sample size, and model complexity grow together, calling for new theoretical tools and perspectives that go beyond traditional asymptotic analyses. In this study, we focus on the learning tasks involving high-dimensional predictors. When SGD is applied directly to such data, the dimensionality of the feature space propagates into the optimization process, resulting in a highdimensional (HD) parameter space. Algorithmically, one trending strategy is to approximate the gradient updates using a low-rank representation to reduce memory costs and accelerate computation (Wang et al., 2018; Vogels et al., 2019; Kozak et al., 2019; Kasiviswanathan, 2021; Zhao et al., 2024). Theoretically, despite the vast literature on SGD, convergence guarantees of HD-SGD remain limited (Garrigos & Gower, 2023; Li et al., 2025).

In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs ($\text{DAGs}$). However, dependency structures can vary across subjects, and ignoring this structural heterogeneity introduces bias and obscures subpopulation-specific dependencies. To address this, we propose Directed Acyclic Graph-based Dependency Clustering via Alternating Direction Method of Multipliers (DAG-DC-ADMM), a unified framework built upon Structural Equation Modeling (SEM) that jointly learns cluster assignments and cluster-specific dependency structures. We encode acyclicity via a smooth constraint and integrate a groupwise truncated Lasso fusion penalty (gTLP) to cluster subjects based on their structural similarity. This yields a nonconvex optimization problem that incorporates sparsity, acyclicity, and structural consensus constraints. We address the nonconvexity by using the augmented Lagrangian method and solve it with an adapted version of the Alternating Direction Method of Multipliers (ADMM) for difference-of-convex programs. For certain graph structures, such as upper triangular adjacency matrices, our algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point. Experiments demonstrate that our method recovers cluster-specific causal dependency structures with a high true positive rate and a low false discovery rate. This capability enables the robust discovery of heterogeneous dependencies across subjects where the subpopulation label is unknown.

Hallucination remains a central failure mode of large language models, but existing benchmarks operationalize it inconsistently across tasks such as summarization, question answering, retrieval-augmented generation, and agentic interaction. This fragmentation makes it unclear whether a mitigation that works in one setting actually reduces hallucinations across contexts. Current hallucination benchmarks either require human annotation and fixed references that may eventually be memorized, or rely on naturalistic observations often recorded in settings that are difficult to reproduce or test systematically. To enable further research on the root causes of hallucination, we introduce HALLUWORLD, an extensible benchmark framework grounded in an explicit reference-world formulation: a model hallucinates when it produces an observable claim that is false with respect to this reference world. Building on this view, we construct a family of synthetic and semi-synthetic benchmark environments in which the reference world is fully specified, the model's observable view is controlled, and hallucination labels can be generated automatically by construction. HALLUWORLD spans multiple settings that are classically representative for AI, i.e., gridworlds, chess, and realistic terminal tasks. This enables controlled variation of key factors such as world complexity, observability, temporal change, and source-conflict policy, allowing us to disentangle hallucinations into more fine-grained error categories. We evaluate frontier and open-weight language models across these settings and find consistent patterns across domains: perceptual hallucination on directly observed information is near-solved for frontier models, while multi-step state tracking and causal forward simulation are still difficult for frontier models, and are not generally solved by extended thinking.

Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM-and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models. Key words: Bessel processes, denoising score matching, diffusion models, empirical Bayes, financial time series, geometric Brownian motion, Tweedie's formula.

We study post-hoc Learning to Defer (L2D) through the lens of ideal distributions: divergence-regularized reweightings of the data distribution under which a model attains low loss. We define deferral via the density-ratio between a model's and an expert's ideals. Using the reduction from density-ratio estimation to class-probability estimation, we derive the DR CPE losses for post-hoc L2D scorers. Deferral decisions are then made by thresholding the scorer, allowing deferral rates to be adjusted without retraining. For KL-based ideal distributions, our deferral rules recovers Chow's rule under the original distribution and a connection to an expert-tilted Bayes posterior -- which incorporates the expert's performance -- depending on if the ideal distributions are joint or marginal distributions. Experimentally, our approach is competitive compared to common baselines and more robust across dataset settings. More broadly, our results cast post-hoc L2D as density-ratio learning between ideal distributions, bridging Chow-style rules, expert comparison, and elucidating connections to related learning settings including anomaly detection.