Signal Estimation with Additive Error Metrics in Compressed Sensing

TL;DR

This paper introduces a versatile algorithm for signal estimation in compressed sensing that minimizes arbitrary error metrics, demonstrating optimality and outperforming existing methods in experiments.

Contribution

The paper presents a general, fast algorithm for minimizing any user-defined error metric in compressed sensing, along with a method to compute theoretical performance limits.

Findings

Algorithm outperforms relaxed BP and CoSaMP in experiments.

Reaches theoretical performance limits for example metrics.

Applicable to arbitrary error metrics in compressed sensing.

Abstract

Compressed sensing typically deals with the estimation of a system input from its noise-corrupted linear measurements, where the number of measurements is smaller than the number of input components. The performance of the estimation process is usually quantified by some standard error metric such as squared error or support set error. In this correspondence, we consider a noisy compressed sensing problem with any arbitrary error metric. We propose a simple, fast, and highly general algorithm that estimates the original signal by minimizing the error metric defined by the user. We verify that our algorithm is optimal owing to the decoupling principle, and we describe a general method to compute the fundamental information-theoretic performance limit for any error metric. We provide two example metrics --- minimum mean absolute error and minimum mean support error --- and give the theoretical performance limits for these two cases. Experimental results show that our algorithm outperforms methods such as relaxed belief propagation (relaxed BP) and compressive sampling matching pursuit (CoSaMP), and reaches the suggested theoretical limits for our two example metrics.