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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. 

Abstract In the form of multidimensional arrays, tensor data have become increasingly prevalent in modern scientific studies and biomedical applications such as computational biology, brain imaging analysis, and process monitoring system. These data are intrinsically heterogeneous with complex dependencies and structure. Therefore, ad‐hoc dimension reduction methods on tensor data may lack statistical efficiency and can obscure essential findings. Model‐based clustering is a cornerstone of multivariate statistics and unsupervised learning; however, existing methods and algorithms are not designed for tensor‐variate samples. In this article, we propose a tensor envelope mixture model (TEMM) for simultaneous clustering and multiway dimension reduction of tensor data. TEMM incorporates tensor‐structure‐preserving dimension reduction into mixture modeling and drastically reduces the number of free parameters and estimative variability. An expectation‐maximization‐type algorithm is developed to obtain likelihood‐based estimators of the cluster means and covariances, which are jointly parameterized and constrained onto a series of lower dimensional subspaces known as the tensor envelopes. We demonstrate the encouraging empirical performance of the proposed method in extensive simulation studies and a real data application in comparison with existing vector and tensor clustering methods. 

The expectation-maximization (EM) algorithm and its variants are widely used in statistics. In high-dimensional mixture linear regression, the model is assumed to be a finite mixture of linear regression and the number of predictors is much larger than the sample size. The standard EM algorithm, which attempts to find the maximum likelihood estimator, becomes infeasible for such model. We devise a group lasso penalized EM algorithm and study its statistical properties. Existing theoretical results of regularized EM algorithms often rely on dividing the sample into many independent batches and employing a fresh batch of sample in each iteration of the algorithm. Our algorithm and theoretical analysis do not require sample-splitting, and can be extended to multivariate response cases. The proposed methods also have encouraging performances in numerical studies.