Reconstruction of a Low-Rank Matrix in the Presence of Gaussian Noise

• Activity Number: 451
• Type: Topic Contributed
• Date/Time: Wednesday, August 1, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Andrey Shabalin*+ and Andrew B Nobel
• Companies: The University of North Carolina at Chapel Hill and The University of North Carolina at Chapel Hill
• Keywords: singular value decomposition ; high-dimensional data ; rank ; random matrix theory ; matrix reconstruction

Abstract

We address the problem of reconstructing a low-rank signal matrix observed with additive Gaussian noise. The common approaches to the matrix reconstruction problem are soft and hard thresholding of the singular values of the observed matrix. However, a reconstruction method does not have to belong to these families of methods, nor even be based on the SVD of the observed matrix. First, we establish that under mild assumptions one can restrict their attention to orthogonally equivariant reconstruction methods (OERM), which act only on the singular values of the observed matrix and do not affect its singular vectors. Thus, we restrict our attention to the OERM family, a superset of all soft and hard thresholding methods. Then, we design a new OERM method based on recent results in random matrix theory. It aims to reverse the effect of the noise on the SVD of the signal matrix. In conjunction with the proposed reconstruction method we introduce a Kolmogorov-Smirnov based estimator of the noise variance. With an extensive simulation study we show that the proposed method outperforms oracle versions of soft and hard thresholding methods, and closely matches the OERM oracle.