OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage

The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix buried in noise by optimal (re)weighting of the singular vectors of the measurement matrix. We exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the i.i.d. Gaussian noise case. Our analysis brings into sharp focus the shrinkage-and-thresholding form of the optimal weights, the non-convex nature of the associated shrinkage function (on the singular values) and explains why matrix regularization via singular value thresholding with convex penalty functions (such as the nuclear norm) will always be suboptimal. We validate our theoretical predictions with numerical simulations, develop an implementable algorithm (OptShrink) that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries.Comment: Published version. The algorithm can be downloaded from http://www.eecs.umich.edu/~rajnrao/optshrin

High Dimensional Low Rank plus Sparse Matrix Decomposition

This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on optimization problems with complexity that scales with the dimension of the data, which limits their scalability. Furthermore, existing randomized approaches mostly rely on uniform random sampling, which is quite inefficient for many real world data matrices that exhibit additional structures (e.g. clustering). In this paper, a scalable subspace-pursuit approach that transforms the decomposition problem to a subspace learning problem is proposed. The decomposition is carried out using a small data sketch formed from sampled columns/rows. Even when the data is sampled uniformly at random, it is shown that the sufficient number of sampled columns/rows is roughly O(r\mu), where \mu is the coherency parameter and r the rank of the low rank component. In addition, adaptive sampling algorithms are proposed to address the problem of column/row sampling from structured data. We provide an analysis of the proposed method with adaptive sampling and show that adaptive sampling makes the required number of sampled columns/rows invariant to the distribution of the data. The proposed approach is amenable to online implementation and an online scheme is proposed.Comment: IEEE Transactions on Signal Processin