TARM: A Turbo-type Algorithm for Affine Rank Minimization

TL;DR

This paper introduces TARM, an iterative turbo-type algorithm for affine rank minimization that efficiently recovers low-rank matrices from noisy measurements, demonstrating faster convergence and superior performance in matrix completion tasks.

Contribution

The paper proposes TARM, a novel message passing-based algorithm for ARM, with proven convergence speed and extended application to matrix completion.

Findings

TARM converges faster than existing algorithms.

State evolution accurately predicts TARM behavior for ROIL measurements.

TARM outperforms counterparts in matrix completion with tuned parameters.

Abstract

The affine rank minimization (ARM) problem arises in many real-world applications. The goal is to recover a low-rank matrix from a small amount of noisy affine measurements. The original problem is NP-hard, and so directly solving the problem is computationally prohibitive. Approximate low-complexity solutions for ARM have recently attracted much research interest. In this paper, we design an iterative algorithm for ARM based on message passing principles. The proposed algorithm is termed turbo-type ARM (TARM), as inspired by the recently developed turbo compressed sensing algorithm for sparse signal recovery. We show that, when the linear operator for measurement is right-orthogonally invariant (ROIL), a scalar function called state evolution can be established to accurately predict the behaviour of the TARM algorithm. We also show that TARM converges much faster than the counterpart algorithms for low-rank matrix recovery. We further extend the TARM algorithm for matrix completion, where the measurement operator corresponds to a random selection matrix. We show that, although the state evolution is not accurate for matrix completion, the TARM algorithm with carefully tuned parameters still significantly outperforms its counterparts.