Optimal convex lifted sparse phase retrieval and PCA with an atomic matrix norm regularizer

TL;DR

This paper introduces a new atomic matrix norm regularizer for convex formulations of sparse phase retrieval and PCA, providing near-optimal guarantees and a practical heuristic algorithm with competitive empirical performance.

Contribution

The paper proposes a novel mixed atomic matrix norm for convex sparse phase retrieval and PCA, with theoretical guarantees and a practical heuristic algorithm.

Findings

Near-optimal sample complexity and error guarantees.

Practical heuristic algorithm performs comparably to state-of-the-art.

New atomic norm promotes low-rank matrices with sparse factors.

Abstract

We present novel analysis and algorithms for solving sparse phase retrieval and sparse principal component analysis (PCA) with convex lifted matrix formulations. The key innovation is a new mixed atomic matrix norm that, when used as regularization, promotes low-rank matrices with sparse factors. We show that convex programs with this atomic norm as a regularizer provide near-optimal sample complexity and error rate guarantees for sparse phase retrieval and sparse PCA. While we do not know how to solve the convex programs exactly with an efficient algorithm, for the phase retrieval case we carefully analyze the program and its dual and thereby derive a practical heuristic algorithm. We show empirically that this practical algorithm performs similarly to existing state-of-the-art algorithms.