Improved Sparse Low-Rank Matrix Estimation

TL;DR

This paper introduces a novel non-convex penalty-based approach for more accurately estimating sparse low-rank matrices from noisy data, outperforming traditional convex methods in applications like audio denoising and biological network analysis.

Contribution

It proposes a parameterized non-convex penalty function that ensures strict convexity and improves sparse low-rank matrix estimation over convex nuclear norm methods.

Findings

Better estimation accuracy than convex methods

Successful application to audio signal denoising

Effective in biological network data reconstruction

Abstract

We address the problem of estimating a sparse low-rank matrix from its noisy observation. We propose an objective function consisting of a data-fidelity term and two parameterized non-convex penalty functions. Further, we show how to set the parameters of the non-convex penalty functions, in order to ensure that the objective function is strictly convex. The proposed objective function better estimates sparse low-rank matrices than a convex method which utilizes the sum of the nuclear norm and the $\ell_1$ norm. We derive an algorithm (as an instance of ADMM) to solve the proposed problem, and guarantee its convergence provided the scalar augmented Lagrangian parameter is set appropriately. We demonstrate the proposed method for denoising an audio signal and an adjacency matrix representing protein interactions in the `Escherichia coli' bacteria.