Lecture notes on non-convex algorithms for low-rank matrix recovery

TL;DR

This paper reviews recent advances in non-convex algorithms for low-rank matrix recovery, highlighting new proof techniques and their applications in signal processing, imaging, and machine learning.

Contribution

It provides an accessible overview of recent theoretical progress and proof methods in non-convex low-rank matrix recovery algorithms.

Findings

Recent non-convex algorithms have shown promising recovery guarantees.

New proof techniques improve understanding of algorithm behavior.

The review bridges theory and practical applications in various fields.

Abstract

Low-rank matrix recovery problems are inverse problems which naturally arise in various fields like signal processing, imaging and machine learning. They are non-convex and NP-hard in full generality. It is therefore a delicate problem to design efficient recovery algorithms and to provide rigorous theoretical insights on the behavior of these algorithms. The goal of these notes is to review recent progress in this direction for the class of so-called "non-convex algorithms", with a particular focus on the proof techniques. Although they aim at presenting very recent research works, these notes have been written with the intent to be, as much as possible, accessible to non-specialists.