Estimating information from structured data is acentral theme in statistics that by now has found applications in a wide array of disciplines.

Traditional data-driven methods, effective for deterministic systems or stochastic differential equations (SDEs) with Gaussian noise, fail to handle the discontinuous sample paths and heavy-tailed fluctuations characteristic of Lévy processes, particularly when the noise is state-dependent. To bridge this gap, we establish nonlocal Kramers-Moyal formulas, rigorously generalizing the classical Kramers-Moyal relations to SDEs with multiplicative Lévy noise. These formulas provide a direct link between short-time transition probability densities (or sample path statistics) and the underlying SDE coefficients: the drift vector, diffusion matrix, Lévy jump measure kernel, and Lévy noise intensity functions. Leveraging these theoretical foundations, we develop novel data-driven algorithms capable of simultaneously identifying all governing components from data and establish convergence results and error analysis for the algorithms. We validate the framework through extensive numerical experiments on prototypical systems. This work provides a principled and practical toolbox for discovering interpretable SDE models governing complex systems influenced by discontinuous, heavy-tailed, state-dependent fluctuations, with broad applicability in climate science, neuroscience, epidemiology, finance, and biological physics.

We study acceleration for distributed sparse regression in {\it high-dimensions}, which allows the parameter size to exceed and grow faster than the sample size. When applicable, existing distributed algorithms employing acceleration perform poorly in this setting, theoretically and numerically. We propose a new accelerated distributed algorithm suitable for high-dimensions. The method couples a suitable instance of accelerated Nesterov's proximal gradient with consensus and gradient-tracking mechanisms, aiming at estimating locally the gradient of the empirical loss while enforcing agreement on the local estimates. Under standard assumptions on the statistical model and tuning parameters, the proposed method is proved to globally converge at {\it linear} rate to an estimate that is within the {\it statistical precision} of the model. The iteration complexity scales as $\mathcal{O}(\sqrt{\kappa})$, while the communications per iteration are at most $\widetilde{\mathcal{O}}(\log m/(1-\rho))$, where $\kappa$ is the restricted condition number of the empirical loss, $m$ is the number of agents, and $\rho\in (0,1)$ measures the network connectivity. As by-product of our design, we also report an accelerated method for high-dimensional estimations over master-worker architectures, which is of independent interest and compares favorably with existing works.

Online sparse linear regression is the task of applying linear regression analysis to examples arriving sequentially subject to a resource constraint that a limited number of features of examples can be observed. Despite its importance in many practical applications, it has been recently shown that there is no polynomial-time sublinear-regret algorithm unless NP BPP, and only an exponential-time sublinear-regret algorithm has been found. In this paper, we introduce mild assumptions to solve the problem.

Most existing approaches to distributed sparse regression assume the data is partitioned by samples.

This article explores novel data-driven modeling approaches for analyzing and approximating the Universal Thermal Climate Index (UTCI), a physiologically-based metric integrating multiple atmospheric variables to assess thermal comfort. Given the nonlinear, multivariate structure of UTCI, we investigate symbolic and sparse regression techniques as tools for interpretable and efficient function approximation. In particular, we highlight the benefits of using orthogonal polynomial bases-such as Legendre polynomials-in sparse regression frameworks, demonstrating their advantages in stability, convergence, and hierarchical interpretability compared to standard polynomial expansions. We demonstrate that our models achieve significantly lower root-mean squared losses than the widely used sixth-degree polynomial benchmark-while using the same or fewer parameters. By leveraging Legendre polynomial bases, we construct models that efficiently populate a Pareto front of accuracy versus complexity and exhibit stable, hierarchical coefficient structures across varying model capacities. Training on just 20% of the data, our models generalize robustly to the remaining 80%, with consistent performance under bootstrapping. The decomposition effectively approximates the UTCI as a Fourier-like expansion in an orthogonal basis, yielding results near the theoretical optimum in the L2 (least squares) sense. We also connect these findings to the broader context of equation discovery in environmental modeling, referencing probabilistic grammar-based methods that enforce domain consistency and compactness in symbolic expressions. Taken together, these results illustrate how combining sparsity, orthogonality, and symbolic structure enables robust, interpretable modeling of complex environmental indices like UTCI - and significantly outperforms the state-of-the-art approximation in both accuracy and efficiency.

We present two main results, both of them follow from a more general statement; see Theorem B.3. Before formally stating the results, we define the model as follows.

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The advent of multi-omics technologies has revolutionized biomedical research, enabling simultaneous interrogation of genomic, transcriptomic, proteomic, and metabolomic layers [Wang et al., 2021a]. This integrative paradigm has yielded unprecedented insights into the molecular architecture of complex diseases, particularly neurodegenerative disorders such as Alzheimer's disease. However, multi-omics datasets are often characterized by high-dimensional variables and limited sample sizes--a configuration known as high-dimension low-sample size (HDLSS). Under such constraints, conventional statistical methods suffer from reduced power and unrealistic assumptions [Fan and Lv, 2008], while deep learning models may exhibit overfitting and lack interpretability [LeCun et al., 2015]. Recent advances in dementia biomarker discovery have embraced multi-omics integration. For example, Iturria-Medina [2018] fused neuroimaging and omics data to identify disease-relevant signatures. Zhang [2020] employed transcriptomic-proteomic fusion to uncover molecular markers, and Lee [2022] demonstrated the discriminative utility of metabolomic features in Alzheimer's pathology. These efforts build upon foundational work in integrative omics [Hasin, 2017, Karczewski and Snyder, 2018], yet challenges persist in elucidating latent gene networks and selecting statistically robust features amidst inter-feature dependencies.