Sublinear-Time Algorithms for Compressive Phase Retrieval

TL;DR

This paper introduces sublinear-time algorithms for the compressive phase retrieval problem, enabling efficient reconstruction of sparse signals from intensity-only measurements using combinatorial methods.

Contribution

It presents the first sublinear-time algorithms for various variants of compressive phase retrieval, employing combinatorial techniques and near-optimal measurements.

Findings

Algorithms operate in sublinear time.

Uses combinatorial techniques for reconstruction.

Achieves near-optimal measurement complexity.

Abstract

In the compressive phase retrieval problem, or phaseless compressed sensing, or compressed sensing from intensity only measurements, the goal is to reconstruct a sparse or approximately $k$-sparse vector $x \in \mathbb{R}^n$ given access to $y= |\Phi x|$, where $|v|$ denotes the vector obtained from taking the absolute value of $v\in\mathbb{R}^n$ coordinate-wise. In this paper we present sublinear-time algorithms for different variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science. Our algorithms use pure combinatorial techniques and near-optimal number of measurements.