Online regularized matrix regression with streaming data

Abstract

As extensions of vector data with ultrahigh dimensionality and complex structures, matrix data are fast emerging in a large variety of scientific applications. In this paper, we consider the matrix regression with streaming data and propose two-stage online regularized estimators with nuclear norm (NN) and adaptive nuclear norm (ANN) penalties, respectively. In the first stage, an equivalent form of offline matrix regression loss function using current raw data and summary statistics from historical data is established. In the second stage, gradient descent algorithm and soft thresholding methods are implemented iteratively to obtain the proposed online NN and ANN estimators. We establish the asymptotic properties of the resulting online regularized estimators and show the rank selection consistency for the online ANN estimator. The finite-sample performance of the proposed estimators is studied through simulations and an application to Beijing Air Quality data set.