Reconstruction of a high-dimensional low-rank matrix

Abstract

We consider the problem of recovering a low-rank signal matrix in high-dimensional situations. The main issue is how to estimate the signal matrix in the presence of huge noise. We introduce the power spiked model to describe the structure of singular values of a huge data matrix. We first consider the conventional PCA to recover the signal matrix and show that the estimation of the signal matrix holds consistency properties under severe conditions. The conventional PCA is heavily subjected to the noise. In order to reduce the noise we apply the noise-reduction (NR) methodology and propose a new estimation of the signal matrix. We show that the proposed estimation by the NR method holds the consistency properties under mild conditions and improves the error rate of the conventional PCA effectively. Finally, we demonstrate the reconstruction procedures by using a microarray data set.