Alternating Least-Squares for Low-Rank Matrix Reconstruction

TL;DR

This paper introduces an iterative alternating least-squares algorithm for reconstructing low-rank matrices from limited measurements, capable of leveraging known matrix structures like Hankel, Toeplitz, and positive semidefinite forms.

Contribution

The paper presents a novel ALS-based method tailored for low-rank matrix reconstruction that exploits specific matrix structures and compares its performance to theoretical bounds.

Findings

Algorithm effectively reconstructs low-rank matrices from undersampled data.

Performance approaches Cramér-Rao bounds in simulations.

Capable of utilizing a-priori structural knowledge of matrices.

Abstract

For reconstruction of low-rank matrices from undersampled measurements, we develop an iterative algorithm based on least-squares estimation. While the algorithm can be used for any low-rank matrix, it is also capable of exploiting a-priori knowledge of matrix structure. In particular, we consider linearly structured matrices, such as Hankel and Toeplitz, as well as positive semidefinite matrices. The performance of the algorithm, referred to as alternating least-squares (ALS), is evaluated by simulations and compared to the Cram\'er-Rao bounds.