ADMiRA: Atomic Decomposition for Minimum Rank Approximation

TL;DR

This paper introduces ADMiRA, an efficient greedy algorithm for low-rank matrix recovery that extends vector-based methods, providing theoretical guarantees and demonstrating competitive performance in matrix completion tasks.

Contribution

It proposes an atomic decomposition framework and an algorithm extending CoSaMP to matrices, with performance guarantees based on the rank-restricted isometry property.

Findings

ADMiRA is efficient and guarantees convergence under certain conditions.

The algorithm performs well in matrix completion even without the ideal measurement operator.

Numerical experiments show ADMiRA's competitiveness in practical scenarios.

Abstract

We address the inverse problem that arises in compressed sensing of a low-rank matrix. Our approach is to pose the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition that provides an analogy between parsimonious representations of a sparse vector and a low-rank matrix. Efficient greedy algorithms to solve the inverse problem for the vector case are extended to the matrix case through this atomic decomposition. In particular, we propose an efficient and guaranteed algorithm named ADMiRA that extends CoSaMP, its analogue for the vector case. The performance guarantee is given in terms of the rank-restricted isometry property and bounds both the number of iterations and the error in the approximate solution for the general case where the solution is approximately low-rank and the measurements are noisy. With a sparse measurement operator such as the one arising in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. The numerical experiments for the matrix completion problem show that, although the measurement operator in this case does not satisfy the rank-restricted isometry property, ADMiRA is a competitive algorithm for matrix completion.