We study rank selection for low-rank tensor regression under random covariates design. Under a Gaussian random-design model and some mild conditions, we derive population expressions for the expected training-testing discrepancy (optimism) for both CP and Tucker decomposition. We further demonstrate that the optimism is minimized at the true tensor rank for both CP and Tucker regression. This yields a prediction-oriented rank-selection rule that aligns with cross-validation and extends naturally to tensor-model averaging. We also discuss conditions under which under- or over-ranked models may appear preferable, thereby clarifying the scope of the method. Finally, we showcase its practical utility on a real-world image regression task and extend its application to tensor-based compression of neural network, highlighting its potential for model selection in deep learning.

Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon and SGD on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and moreover Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of Muon and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.

Missing data is a ubiquitous challenge in data analysis, often leading to biased and inaccurate results. Traditional imputation methods usually assume that the missingness mechanism is missing-at-random (MAR), where the missingness is independent of the missing values themselves. This assumption is frequently violated in real-world scenarios, prompted by recent advances in imputation methods using deep learning to address this challenge. However, these methods neglect the crucial issue of nonparametric identifiability in missing-not-at-random (MNAR) data, which can lead to biased and unreliable results. This paper seeks to bridge this gap by proposing a novel framework based on deep latent variable models for MNAR data. Building on the assumption of conditional no self-censoring given latent variables, we establish the identifiability of the data distribution. This crucial theoretical result guarantees the feasibility of our approach. To effectively estimate unknown parameters, we develop an efficient algorithm utilizing importance-weighted autoencoders. We demonstrate, both theoretically and empirically, that our estimation process accurately recovers the ground-truth joint distribution under specific regularity conditions. Extensive simulation studies and real-world data experiments showcase the advantages of our proposed method compared to various classical and state-of-the-art approaches to missing data imputation.

Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a way to incorporate topological priors or to regularize machine learning models. This is usually achieved by minimizing adequate, topologically-informed losses based on these descriptors, which, in turn, naturally raises theoretical and practical questions about the possibility of optimizing such loss functions using gradient-based algorithms. This has been an active research field in the topological data analysis community over the last decade, and various techniques have been developed to enable optimization of persistence-based loss functions with gradient descent schemes. This survey presents the current state of this field, covering its theoretical foundations, the algorithmic aspects, and showcasing practical uses in several applications. It includes a detailed introduction to persistence theory and, as such, aims at being accessible to mathematicians and data scientists newcomers to the field. It is accompanied by an open-source library which implements the different approaches covered in this survey, providing a convenient playground for researchers to get familiar with the field.

We study the unconstrained minimization of a smooth and strongly convex population loss function under a stochastic oracle that introduces both additive and multiplicative noise; this is a canonical and widely-studied setting that arises across operations research, signal processing, and machine learning. We begin by showing that standard approaches such as sample average approximation and robust (or averaged) stochastic approximation can lead to suboptimal -- and in some cases arbitrarily poor -- performance with realistic finite sample sizes. In contrast, we demonstrate that a carefully designed variance reduction strategy, which we term VISOR for short, can significantly outperform these approaches while using the same sample size. Our upper bounds are complemented by finite-sample, information-theoretic local minimax lower bounds, which highlight fundamental, instance-dependent factors that govern the performance of any estimator. Taken together, these results demonstrate that an accelerated variant of VISOR is instance-optimal, achieving the best possible sample complexity up to logarithmic factors while also attaining optimal oracle complexity. We apply our theory to generalized linear models and improve upon classical results. In particular, we obtain the best-known non-asymptotic, instance-dependent generalization error bounds for stochastic methods, even in linear regression.

Tensors provide a structured representation for multidimensional data, yet discretization can obscure important information when such data originates from continuous processes. We address this limitation by introducing a functional Tucker decomposition (FTD) that embeds mode-wise continuity constraints directly into the decomposition. The FTD employs reproducing kernel Hilbert spaces (RKHS) to model continuous modes without requiring an a-priori basis, while preserving the multi-linear subspace structure of the Tucker model. Through RKHS-driven representation, the model yields adaptive and expressive factor descriptions that enable targeted modeling of subspaces. The value of this approach is demonstrated in domain-variant tensor classification. In particular, we illustrate its effectiveness with classification tasks in hyperspectral imaging and multivariate time series analysis, highlighting the benefits of combining structural decomposition with functional adaptability.

We study statistical estimation in a student--teacher setting, where predictions from a pre-trained teacher are used to guide a student model. A standard approach is to train the student to directly match the teacher's outputs, which we refer to as student soft matching (SM). This approach directly propagates any systematic bias or mis-specification present in the teacher, thereby degrading the student's predictions. We propose and analyze an alternative scheme, known as residual-as-teacher (RaT), in which the teacher is used to estimate residuals in the student's predictions. Our analysis shows how the student can thereby emulate a proximal gradient scheme for solving an oracle optimization problem, and this provably reduces the effect of teacher bias. For general student--teacher pairs, we establish non-asymptotic excess risk bounds for any RaT fixed point, along with convergence guarantees for the student-teacher iterative scheme. For kernel-based student--teacher pairs, we prove a sharp separation: the RaT method achieves the minimax-optimal rate, while the SM method incurs constant prediction error for any sample size. Experiments on both synthetic data and ImageNette classification under covariate shift corroborate our theoretical findings.

Causal representation learning seeks to uncover causal relationships among high-level latent variables from low-level, entangled, and noisy observations. Existing approaches often either rely on deep neural networks, which lack interpretability and formal guarantees, or impose restrictive assumptions like linearity, continuous-only observations, and strong structural priors. These limitations particularly challenge applications with a large number of discrete latent variables and mixed-type observations. To address these challenges, we propose discrete causal representation learning (DCRL), a generative framework that models a directed acyclic graph among discrete latent variables, along with a sparse bipartite graph linking latent and observed layers. This design accommodates continuous, count, and binary responses through flexible measurement models while maintaining interpretability. Under mild conditions, we prove that both the bipartite measurement graph and the latent causal graph are identifiable from the observed data distribution alone. We further propose a three-stage estimate-resample-discovery pipeline: penalized estimation of the generative model parameters, resampling of latent configurations from the fitted model, and score-based causal discovery on the resampled latents. We establish the consistency of this procedure, ensuring reliable recovery of the latent causal structure. Empirical studies on educational assessment and synthetic image data demonstrate that DCRL recovers sparse and interpretable latent causal structures.

We propose a method for constructing distribution-free prediction intervals in nonparametric instrumental variable regression (NPIV), with finite-sample coverage guarantees. Building on the conditional guarantee framework in conformal inference, we reformulate conditional coverage as marginal coverage over a class of IV shifts $\mathcal{F}$. Our method can be combined with any NPIV estimator, including sieve 2SLS and other machine-learning-based NPIV methods such as neural networks minimax approaches. Our theoretical analysis establishes distribution-free, finite-sample coverage over a practitioner-chosen class of IV shifts.

The theory of Local Intrinsic Dimensionality (LID) has become a valuable tool for characterizing local complexity within and across data manifolds, supporting a range of data mining and machine learning tasks. Accurate LID estimation requires samples drawn from small neighborhoods around each query to avoid biases from nonlocal effects and potential manifold mixing, yet limited data within such neighborhoods tends to cause high estimation variance. As a variance reduction strategy, we propose an ensemble approach that uses subbagging to preserve the local distribution of nearest neighbor (NN) distances. The main challenge is that the uniform reduction in total sample size within each subsample increases the proximity threshold for finding a fixed number k of NNs around the query. As a result, in the specific context of LID estimation, the sampling rate has an additional, complex interplay with the neighborhood size, where both combined determine the sample size as well as the locality and resolution considered for estimation. We analyze both theoretically and experimentally how the choice of the sampling rate and the k-NN size used for LID estimation, alongside the ensemble size, affects performance, enabling informed prior selection of these hyper-parameters depending on application-based preferences. Our results indicate that within broad and well-characterized regions of the hyper-parameters space, using a bagged estimator will most often significantly reduce variance as well as the mean squared error when compared to the corresponding non-bagged baseline, with controllable impact on bias. We additionally propose and evaluate different ways of combining bagging with neighborhood smoothing for substantial further improvements on LID estimation performance.