Multiple Regression for Matrix and Vector Predictors: Models, Theory, Algorithms, and Beyond

Abstract

Matrix regression plays an important role in modern data analysis due to its ability to handle complex relationships involving both matrix and vector variables. We propose a class of regularized regression models capable of predicting both matrix and vector variables, accommodating various regularization techniques tailored to the inherent structures of the data. We establish the consistency of our estimator when penalizing the nuclear norm of the matrix variable and the ${\ell }_1$ norm of the vector variable. To tackle the general regularized regression model, we propose a unified framework based on an efficient preconditioned proximal point algorithm. Numerical experiments demonstrate the superior estimation and prediction accuracy of our proposed estimator, as well as the efficiency of our algorithm compared to the state-of-the-art solvers.

References in corpus (7)

• A note on the group lasso and a sparse group lasso
• Support union recovery in high-dimensional multivariate regression
• SLOPE - Adaptive variable selection via convex optimization
• An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems
• A sparse semismooth Newton based proximal majorization-minimization algorithm for nonconvex square-root-loss regression problems
• Solving the OSCAR and SLOPE Models Using a Semismooth Newton-Based Augmented Lagrangian Method
• A Highly Efficient Algorithm for Solving Exclusive Lasso Problems