One-Bit Compressive Sensing of Dictionary-Sparse Signals

TL;DR

This paper extends one-bit compressive sensing to signals sparse in overcomplete dictionaries, providing theoretical guarantees for recovery using convex and thresholding algorithms under Gaussian sensing matrices.

Contribution

It introduces a unified theory for one-bit compressive sensing with dictionary-sparse signals, broadening the scope beyond basis-sparse signals.

Findings

Algorithms can recover dictionary-sparse signals with high probability

Gaussian sensing matrices satisfy natural assumptions for recovery

Convex and thresholding methods are effective in this setting

Abstract

One-bit compressive sensing has extended the scope of sparse recovery by showing that sparse signals can be accurately reconstructed even when their linear measurements are subject to the extreme quantization scenario of binary samples---only the sign of each linear measurement is maintained. Existing results in one-bit compressive sensing rely on the assumption that the signals of interest are sparse in some fixed orthonormal basis. However, in most practical applications, signals are sparse with respect to an overcomplete dictionary, rather than a basis. There has already been a surge of activity to obtain recovery guarantees under such a generalized sparsity model in the classical compressive sensing setting. Here, we extend the one-bit framework to this important model, providing a unified theory of one-bit compressive sensing under dictionary sparsity. Specifically, we analyze several different algorithms---based on convex programming and on hard thresholding---and show that, under natural assumptions on the sensing matrix (satisfied by Gaussian matrices), these algorithms can efficiently recover analysis-dictionary-sparse signals in the one-bit model.