Robust Approaches for Matrix-Valued Parameters

Abstract

Modern large data sets inevitably contain outliers that deviate from the model assumptions. However, many widely used estimators, such as maximum likelihood estimators and least squared estimators, perform weakly with the existence of outliers. Alternatively, many statistical modeling approaches have matrices as the parameters. We consider penalized estimators for matrix-valued parameters with a focus on their robustness properties in the presence of outliers. We propose a general framework for robust modeling with matrix-valued parameters by minimizing robust loss functions with penalization. However, there are challenges to this approach in both computation and theoretical analysis. To tackle the computational challenges from the large size of the data, non-smoothness of robust loss functions, and the slow speed of matrix operations, we propose to apply the Frank-Wolfe algorithm, a first-order algorithm for optimization on a restricted region with low computation burden per iteration. Theoretically, we establish finite-sample error bounds under high-dimensional settings. We show that the estimation errors are bounded by small terms and converge in probability to zero under mild conditions in a neighborhood of the true model. Our method accommodates a broad classes of modeling problems using robust loss functions with penalization. Concretely, we study three cases: matrix completion, multivariate regression, and network estimation. For all cases, we illustrate the robustness of the proposed method both theoretically and numerically.