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The problem of classifying multiple categorical responses is fundamental in modern machine learning and statistics, with diverse applications in fields such as bioinformatics and imaging. This manuscript investigates linear discriminant analysis (LDA) with high-dimensional predictors and multiple multi-class responses. Specifically, we first examine two different classification scenarios under the bivariate LDA model: joint classification of the two responses and conditional classification of one response while observing the other. To achieve optimal classification rules for both scenarios, we introduce two novel tensor formulations of the discriminant coefficients and corresponding regularization strategies. For joint classification, we propose an overlapping group lasso penalty and a blockwise coordinate descent algorithm to efficiently compute the joint discriminant coefficient tensors. For conditional classification, we utilize an alternating direction method of multipliers (ADMM) algorithm to compute the discriminant coefficient tensors under new constraints. We then extend our method and algorithms to general multivariate responses. Finally, we validate the effectiveness of our approach through simulation studies and applications to benchmark datasets. 

ABSTRACT We address the challenge of estimating regression coefficients and selecting relevant predictors in the context of mixed linear regression in high dimensions, where the number of predictors greatly exceeds the sample size. Recent advancements in this field have centered on incorporating sparsity-inducing penalties into the expectation-maximization (EM) algorithm, which seeks to maximize the conditional likelihood of the response given the predictors. However, existing procedures often treat predictors as fixed or overlook their inherent variability. In this paper, we leverage the independence between the predictor and the latent indicator variable of mixtures to facilitate efficient computation and also achieve synergistic variable selection across all mixture components. We establish the non-asymptotic convergence rate of the proposed fast group-penalized EM estimator to the true regression parameters. The effectiveness of our method is demonstrated through extensive simulations and an application to the Cancer Cell Line Encyclopedia dataset for the prediction of anticancer drug sensitivity. 

The ratio of atmosphere-derived¹⁰Be to continent-derived⁹Be in marine sediments has been used to probe the long-term relationship between continental denudation and climate. However, its application is complicated by uncertainty in⁹Be transfer through the land-ocean interface. The riverine dissolved load alone is insufficient to close the marine⁹Be budget, largely due to substantial removal of riverine⁹Be to continental margin sediments. We focus on the ultimate fate of this latter Be. We present sediment pore-water Be profiles from diverse continental margin environments to quantify the diagenetic Be release to the ocean. Our results suggest that pore-water Be cycling is mainly controlled by particulate supply and Mn-Fe cycling, leading to higher benthic fluxes on shelves. Benthic fluxes may help close the⁹Be budget and are at least comparable to, or higher (~2-fold) than, the riverine dissolved input. These observations demand a revised model framework, which considers the potentially dominant benthic source, to robustly interpret marine Be isotopic records.