Statistical mechanics approach to 1-bit compressed sensing

TL;DR

This paper analyzes the performance of L1-norm based recovery in 1-bit compressed sensing using statistical mechanics, revealing the existence of many local optima and proposing a new cavity-inspired algorithm with improved performance.

Contribution

It applies statistical mechanics methods to analyze 1-bit compressed sensing, and introduces a new cavity-inspired recovery algorithm with better performance at lower computational cost.

Findings

Replica method predictions align with experimental results.

Many local optima exist with similar recovery accuracy.

New algorithm outperforms previous methods for dense signals.

Abstract

Compressed sensing is a technique for recovering a high-dimensional signal from lower-dimensional data, whose components represent partial information about the signal, utilizing prior knowledge on the sparsity of the signal. For further reducing the data size of the compressed expression, a scheme to recover the original signal utilizing only the sign of each entry of the linearly transformed vector was recently proposed. This approach is often termed the 1-bit compressed sensing. Here we analyze the typical performance of an L1-norm based signal recovery scheme for the 1-bit compressed sensing using statistical mechanics methods. We show that the signal recovery performance predicted by the replica method under the replica symmetric ansatz, which turns out to be locally unstable for modes breaking the replica symmetry, is in a good consistency with experimental results of an approximate recovery algorithm developed earlier. This suggests that the L1-based recovery problem typically has many local optima of a similar recovery accuracy, which can be achieved by the approximate algorithm. We also develop another approximate recovery algorithm inspired by the cavity method. Numerical experiments show that when the density of nonzero entries in the original signal is relatively large the new algorithm offers better performance than the abovementioned scheme and does so with a lower computational cost.