Sketching low-rank matrices with a shared column space by convex programming

TL;DR

This paper introduces a convex method for jointly estimating multiple low-rank matrices sharing a common column space from noisy measurements, demonstrating improved recovery performance over individual methods.

Contribution

The paper proposes a novel convex estimator regularized by specific matrix norms for joint low-rank matrix recovery with shared column space, including an efficient ADMM algorithm.

Findings

Joint estimation outperforms individual recovery in large-scale settings.

The convex estimator is effectively implemented as a semidefinite program.

Simulation results show improved recovery accuracy compared to existing methods.

Abstract

In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of low-spikiness in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the joint estimation provides a significant gain over the individual recovery of each matrix when the number of matrices sharing a column space and the ambient dimension of the shared column space are large relative to the number of columns in each matrix. The convex estimator is cast as a semidefinite program and an efficient ADMM algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.