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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. 

Tensor regression analysis is finding vast emerging applications in a variety of clinical settings, including neuroimaging, genomics, and dental medicine. The motivation for this paper is a study of periodontal disease (PD) with an order‐3 tensor response: multiple biomarkers measured at prespecified tooth–sites within each tooth, for each participant. A careful investigation would reveal considerable skewness in the responses, in addition to response missingness. To mitigate the shortcomings of existing analysis tools, we propose a new Bayesian tensor response regression method that facilitates interpretation of covariate effects on both marginal and joint distributions of highly skewed tensor responses, and accommodates missing‐at‐random responses under a closure property of our tensor model. Furthermore, we present a prudent evaluation of the overall covariate effects while identifying their possible variations on only a sparse subset of the tensor components. Our method promises Markov chain Monte Carlo (MCMC) tools that are readily implementable. We illustrate substantial advantages of our proposal over existing methods via simulation studies and application to a real data set derived from a clinical study of PD.