Provable Sample-Efficient Sparse Phase Retrieval Initialized by Truncated Power Method

TL;DR

This paper introduces a truncated power method for sparse phase retrieval that reduces the measurement requirements for initialization, achieving near-optimal sample complexity and improving efficiency over existing spectral methods.

Contribution

The paper proposes a novel truncated power method for initialization in sparse phase retrieval, lowering the measurement threshold from quadratic to near-linear in sparsity.

Findings

The method achieves successful initialization with $m=\Omega(\bar{s} s \log n)$ measurements.

When the signal has very few significant components, the sample complexity reduces to $m=\Omega(s \log n)$, which is optimal.

Numerical experiments confirm the method's superior sample efficiency compared to existing algorithms.

Abstract

We study the sparse phase retrieval problem, recovering an $s$-sparse length-$n$ signal from $m$ magnitude-only measurements. Two-stage non-convex approaches have drawn much attention in recent studies for this problem. Despite non-convexity, many two-stage algorithms provably converge to the underlying solution linearly when appropriately initialized. However, in terms of sample complexity, the bottleneck of those algorithms often comes from the initialization stage. Although the refinement stage usually needs only $m=\Omega(s\log n)$ measurements, the widely used spectral initialization in the initialization stage requires $m=\Omega(s^2\log n)$ measurements to produce a desired initial guess, which causes the total sample complexity order-wisely more than necessary. To reduce the number of measurements, we propose a truncated power method to replace the spectral initialization for non-convex sparse phase retrieval algorithms. We prove that $m=\Omega(\bar{s} s\log n)$ measurements, where $\bar{s}$ is the stable sparsity of the underlying signal, are sufficient to produce a desired initial guess. When the underlying signal contains only very few significant components, the sample complexity of the proposed algorithm is $m=\Omega(s\log n)$ and optimal. Numerical experiments illustrate that the proposed method is more sample-efficient than state-of-the-art algorithms.