Covariance matrix estimation in high dimensions is a fundamental problem in machine learning and signal processing. A common structural assumption used to mitigate the challenges posed by high dimensionality is sparsity, which posits that most variable pairs exhibit negligible correlations. In this paper, we revisit the classical problem of positive definite sparse covariance estimation (PDSCE) introduced by Rothman (2012). Unlike many earlier approaches, this formulation incorporates a logarithmic barrier, which guarantees that the resulting covariance estimator is positive definite and thereby ensures the well-posedness of the estimation problem. However, the inclusion of the logarithmic barrier also leads to nontrivial optimization difficulties. To overcome these difficulties, we propose a dual proximal gradient method (DPGM) for solving the PDSCE problem. In contrast to existing primal-space approaches, DPGM operates directly in the dual space. This dual perspective provides several key advantages. First, DPGM significantly reduces computational costs, because positive definiteness is preserved automatically and no iterative subproblem solvers are required. Second, compared with primal optimization algorithms, DPGM offers stronger theoretical guarantees, including principled step size selection and improved iteration complexity. Extensive numerical experiments demonstrate that DPGM consistently outperforms existing methods, which confirms its effectiveness and scalability for high-dimensional sparse covariance estimation.

Covariance matrix estimation is a fundamental problem in multivariate data analysis, which becomes particularly challenging in high-dimensional settings due to the curse of dimensionality. To enhance estimation accuracy, structural regularization is often imposed on the precision matrix (the inverse covariance matrix) for covariance selection. In this paper, we study covariance selection in a distributed setting, where data is spread across a network of agents. We formulate the problem as a Gaussian maximum likelihood estimation problem with structural penalties and propose a novel algorithmic framework called NetGGM. Unlike existing methods that rely on a central coordinator, NetGGM operates in a fully decentralized manner with low computational complexity. We provide theoretical guarantees showing that NetGGM converges linearly to the global optimum while ensuring consensus among agents. Numerical experiments validate its convergence properties and demonstrate that it outperforms state-of-the-art methods in precision matrix estimation.

Large precision matrix estimation is a crucial problem in high-dimensional multivariate data analytics. When the data exhibit heavy-tailedness or outliers, existing estimators based on Gaussian distribution assumption become inefficient. In this paper, we introduce tLasso, a novel adaptive model for robust precision matrix estimation based on an$\ell_{1}$-penalized t-distribution log-likelihood function. We assume the degree of freedom parameter of the t-distribution to be unknown a priori, adapting to the arbitrary heavy-tailedness of data distributions. Then we propose a regularized MCECM algorithm to solve the non-convex estimation problem. Theoretically, we prove the non-asymptotic bound on the estimation error, which establishes the computational and statistical guarantees of the algorithm. Numerical simulations validate the effectiveness of the proposed estimator.

Quantifying aleatoric uncertainty is a challenging task in machine learning. It is important for decision making associated with data-dependent uncertainty in model outcomes. Recently, many empirical studies in modeling aleatoric uncertainty under regression settings primarily rely on either a Gaussian likelihood or moment matching. However, the performance of these methods varies for different datasets whereas discussions on their theoretical guarantees are lacking. In this work, we investigate theoretical aspects of these approaches and establish risk bounds for their estimates. We provide conditions that are sufficient to guarantee the PAC-learnablility of the aleatoric uncertainty. The study suggests that the likelihood and moment matching-based methods enjoy different types of guarantee in their risk bounds, i.e., they calibrate different aspects of the uncertainty and thus exhibit distinct properties in different regimes of the parameter space. Finally, we conduct empirical study which shows promising results and supports our theorems.

The Schrödinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized optimal transport problem, which conducts projections onto every other marginal alternatingly. However, in practice, only approximated projections are accessible and their convergence is not well understood. To fill this gap, we present a first convergence analysis of the Schrödinger bridge algorithm based on approximated projections. As for its practical applications, we apply SBP to probabilistic time series imputation by generating missing values conditioned on observed data. We show that optimizing the transport cost improves the performance and the proposed algorithm achieves the state-of-the-art result in healthcare and environmental data while exhibiting the advantage of exploring both temporal and feature patterns in probabilistic time series imputation.

Many kinds of simultaneously-observed event sequences exhibit mutually exciting or inhibiting patterns. Reliable detection of such temporal dependencies is crucial for scientific investigation. A common model is the Multivariate Hawkes Process (MHP), whose impact function naturally encodes a causal structure in Granger causality. However, the vast majority of existing methods use a transformed standard MHP intensity with a constant baseline, which may be inconsistent with real-world data. On the other hand, modeling irregular and unknown background dynamics directly is a challenge, as one struggles to distinguish the effect of mutual interaction from that of fluctuations in background dynamics. In this paper, we address the short-term temporal dependency detection issue. We show that maximum likelihood estimation (MLE) for cross-impact from MHP has an error that can not be eliminated, but may be reduced by an order of magnitude using a heterogeneous intensity not for the target HP but for the interacting HP. Then we propose a robust and computationally-efficient modification of MLE that does not rely on the prior estimation of the heterogeneous intensity and is thus applicable in a data-limited regime (e.g., few-shot, unrepeated observations). Extensive experiments on various datasets show that our method outperforms existing ones by notable margins, with highlighted novel applications in neuroscience.

Information-theoretic clustering is one of the most promising and principled approaches to finding clusters with minimal apriori assumptions. The key criterion therein is to maximize the mutual information between the data points and their cluster labels. Such an approach, however, does not explicitly promote any type of inter-cluster behavior. We instead propose to maximize the Kullback-Leibler divergence between the underlying data distributions associated to clusters (referred to as cluster distributions). We show it to entail the mutual information criterion along with maximizing cross entropy between the cluster distributions. For practical efficiency, we propose to empirically estimate the objective of KL-D between clusters in its dual form leveraging deep neural nets as a dual function approximator. Remarkably, our theoretical analysis establishes that estimating the divergence measure in its dual form simplifies the problem of clustering to one of optimally finding k-1 cut points for k clusters in the 1-D dual functional space. Overall, our approach enables linear-time clustering algorithms with theoretical guarantees of near-optimality, owing to the submodularity of the objective. We show the empirical superiority of our approach w.r.t. current state-of-the-art methods on the challenging task of clustering noisy timeseries as observed in domains such as neuroscience, healthcare, financial markets, spatio-temporal environmental dynamics, etc.

Evaluating rare but high-stakes events is one of the main challenges in obtaining reliable reinforcement learning policies, especially in large or infinite state/action spaces where limited scalability dictates a prohibitively large number of testing iterations. On the other hand, a biased or inaccurate policy evaluation in a safety-critical system could potentially cause unexpected catastrophic failures during deployment. This paper proposes the Accelerated Policy Evaluation (APE) method, which simultaneously uncovers rare events and estimates the rare event probability in Markov decision processes. The APE method treats the environment nature as an adversarial agent and learns towards, through adaptive importance sampling, the zero-variance sampling distribution for the policy evaluation. Moreover, APE is scalable to large discrete or continuous spaces by incorporating function approximators. We investigate the convergence property of APE in the tabular setting. Our empirical studies show that APE can estimate the rare event probability with a smaller bias while only using orders of magnitude fewer samples than baselines in multi-agent and single-agent environments.

We develop a model-free reinforcement learning approach to solve constrained Markov decision processes, where the objective and budget constraints are in the form of infinite-horizon discounted expectations, and the rewards and costs are learned sequentially from data. We propose a two-stage procedure where we first search over deterministic policies, followed by an aggregation with a mixture parameter search, that generates policies with simultaneous guarantees on near-optimality and feasibility. We also numerically illustrate our approach by applying it to an online advertising problem.

In many learning problems, the training and testing data follow different distributions and a particularly common situation is the \textit{covariate shift}. To correct for sampling biases, most approaches, including the popular kernel mean matching (KMM), focus on estimating the importance weights between the two distributions. Reweighting-based methods, however, are exposed to high variance when the distributional discrepancy is large. On the other hand, the alternate approach of using nonparametric regression (NR) incurs high bias when the training size is limited. In this paper, we propose and analyze a new estimator that systematically integrates the residuals of NR with KMM reweighting, based on a control-variate perspective. The proposed estimator can be shown to either strictly outperform or match the best-known existing rates for both KMM and NR, and thus is a robust combination of both estimators. The experiments shows the estimator works well in practice.