With the rapid development of web services, large amounts of time series data are generated and accumulated across various domains such as finance, healthcare, and online platforms. As such data often co-evolves with multiple variables interacting with each other, estimating the time-varying dependencies between variables (i.e., the dynamic network structure) has become crucial for accurate modeling. However, real-world data is often represented as tensor time series with multiple modes, resulting in large, entangled networks that are hard to interpret and computationally intensive to estimate. In this paper, we propose Kronecker Time-Varying Graphical Lasso (KTVGL), a method designed for modeling tensor time series. Our approach estimates mode-specific dynamic networks in a Kronecker product form, thereby avoiding overly complex entangled structures and producing interpretable modeling results. Moreover, the partitioned network structure prevents the exponential growth of computational time with data dimension. In addition, our method can be extended to stream algorithms, making the computational time independent of the sequence length. Experiments on synthetic data show that the proposed method achieves higher edge estimation accuracy than existing methods while requiring less computation time. To further demonstrate its practical value, we also present a case study using real-world data. Our source code and datasets are available at https://github.com/Higashiguchi-Shingo/KTVGL.

Selecting interpretable feature sets in underdetermined ($n \ll p$) and highly correlated regimes constitutes a fundamental challenge in data science, particularly when analyzing physical measurements. In such settings, multiple distinct sparse subsets may explain the response equally well. Identifying these alternatives is crucial for generating domain-specific insights into the underlying mechanisms, yet conventional methods typically isolate a single solution, obscuring the full spectrum of plausible explanations. We present GEMSS (Gaussian Ensemble for Multiple Sparse Solutions), a variational Bayesian framework specifically designed to simultaneously discover multiple, diverse sparse feature combinations. The method employs a structured spike-and-slab prior for sparsity, a mixture of Gaussians to approximate the intractable multimodal posterior, and a Jaccard-based penalty to further control solution diversity. Unlike sequential greedy approaches, GEMSS optimizes the entire ensemble of solutions within a single objective function via stochastic gradient descent. The method is validated on a comprehensive benchmark comprising 128 synthetic experiments across classification and regression tasks. Results demonstrate that GEMSS scales effectively to high-dimensional settings ($p=5000$) with sample size as small as $n = 50$, generalizes seamlessly to continuous targets, handles missing data natively, and exhibits remarkable robustness to class imbalance and Gaussian noise. GEMSS is available as a Python package 'gemss' at PyPI. The full GitHub repository at https://github.com/kat-er-ina/gemss/ also includes a free, easy-to-use application suitable for non-coders.

Estimating the dimensionality of the latent representation needed for prediction-- the task-relevant dimension--is a difficult, largely unsolved problem with broad scientific applications. We cast it as an Information Bottleneck question: what embedding bottleneck dimension is sufficient to compress predictor and predicted views while preserving their mutual information (MI). We show that standard neural estimators with separable/bilinear critics systematically inflate the inferred dimension, and we address this by introducing a hybrid critic that retains an explicit dimensional bottleneck while allowing flexible nonlinear cross-view interactions, thereby preserving the latent geometry. We further propose a one-shot protocol that reads off the effective dimension from a single over-parameterized hybrid model, without sweeping over bottleneck sizes. We validate the approach on synthetic problems with known task-relevant dimension. We extend the approach to intrinsic dimensionality by constructing paired views of a single dataset, enabling comparison with classical geometric dimension estimators. In noisy regimes where those estimators degrade, our approach remains reliable. Finally, we demonstrate the utility of the method on multiple physics datasets. Before "low-dimensional latent embeddings" became a rallying cry of AI, they were already a basic aim of science: identify a low-dimensional state--a small set of degrees of freedom constructed from observations--that suffices to predict the quantities of interest. The long road from Aristotelian to Newtonian mechanics illustrates that determining the number of such state variables--the relevant latent dimensionality--can be hard, even before one argues about the right variables or the laws that relate them.

Choosing relevant predictors is central to the analysis of biomedical time-to-event data. Classical frequentist inference, however, presumes that the set of covariates is fixed in advance and does not account for data-driven variable selection. As a consequence, naive post-selection inference may be biased and misleading. In right-censored survival settings, these issues may be further exacerbated by the additional uncertainty induced by censoring. We investigate several inference procedures applied after variable selection for the coefficients of the Lasso and its extension, the adaptive Lasso, in the context of the Cox model. The methods considered include sample splitting, exact post-selection inference, and the debiased Lasso. Their performance is examined in a neutral simulation study reflecting realistic covariate structures and censoring rates commonly encountered in biomedical applications. To complement the simulation results, we illustrate the practical behavior of these procedures in an applied example using a publicly available survival dataset.

We study the sample complexity of the best-case Empirical Risk Minimizer in the setting of stochastic convex optimization. We show that there exists an instance in which the sample size is linear in the dimension, learning is possible, but the Empirical Risk Minimizer is likely to be unique and to overfit. This resolves an open question by Feldman. We also extend this to approximate ERMs. Building on our construction we also show that (constrained) Gradient Descent potentially overfits when horizon and learning rate grow w.r.t sample size. Specifically we provide a novel generalization lower bound of $Ω\left(\sqrt{ηT/m^{1.5}}\right)$ for Gradient Descent, where $η$ is the learning rate, $T$ is the horizon and $m$ is the sample size. This narrows down, exponentially, the gap between the best known upper bound of $O(ηT/m)$ and existing lower bounds from previous constructions.

Semi-supervised learning (SSL) addresses the critical challenge of training accurate models when labeled data is scarce but unlabeled data is abundant. Graph-based SSL (GSSL) has emerged as a popular framework that captures data structure through graph representations. Classic graph SSL methods, such as Label Propagation and Label Spreading, aim to compute low-dimensional representations where points with the same labels are close in representation space. Although often effective, these methods can be suboptimal on data with complex label distributions. In our work, we develop AUC-spec, a graph approach that computes a low-dimensional representation that maximizes class separation. We compute this representation by optimizing the Area Under the ROC Curve (AUC) as estimated via the labeled points. We provide a detailed analysis of our approach under a product-of-manifold model, and show that the required number of labeled points for AUC-spec is polynomial in the model parameters. Empirically, we show that AUC-spec balances class separation with graph smoothness. It demonstrates competitive results on synthetic and real-world datasets while maintaining computational efficiency comparable to the field's classic and state-of-the-art methods.

Learning discrete neural samplers is challenging due to the lack of gradients and combinatorial complexity. While stochastic optimal control (SOC) and Schrödinger bridge (SB) provide principled solutions, efficient SOC solvers like adjoint matching (AM), which excel in continuous domains, remain unexplored for discrete spaces. We bridge this gap by revealing that the core mechanism of AM is $\mathit{state}\text{-}\mathit{space~agnostic}$, and introduce $\mathbf{discrete~ASBS}$, a unified framework that extends AM and adjoint Schrödinger bridge sampler (ASBS) to discrete spaces. Theoretically, we analyze the optimality conditions of the discrete SB problem and its connection to SOC, identifying a necessary cyclic group structure on the state space to enable this extension. Empirically, discrete ASBS achieves competitive sample quality with significant advantages in training efficiency and scalability.

Inference on the conditional mean function (CMF) is central to tasks from adaptive experimentation to optimal treatment assignment and algorithmic fairness auditing. In this work, we provide a novel asymptotic anytime-valid test for a CMF global null (e.g., that all conditional means are zero) and contrasts between CMFs, enabling experimenters to make high confidence decisions at any time during the experiment beyond a minimum sample size. We provide mild conditions under which our tests achieve (i) asymptotic type-I error guarantees, (i) power one, and, unlike past tests, (iii) optimal sample complexity relative to a Gaussian location testing. By inverting our tests, we show how to construct function-valued asymptotic confidence sequences for the CMF and contrasts thereof. Experiments on both synthetic and real-world data show our method is well-powered across various distributions while preserving the nominal error rate under continuous monitoring.