Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation

TL;DR

This paper introduces a universal compressed sensing approach that estimates signals without prior knowledge of their structure by leveraging complexity-based priors, supported by theoretical and experimental validation.

Contribution

It develops a universal MAP estimation framework inspired by Kolmogorov complexity, capable of recovering signals with unknown structures from linear measurements.

Findings

The proposed method achieves comparable or better reconstruction quality than existing algorithms.

It demonstrates effectiveness especially for low-complexity sources lacking sparsity.

Theoretical analysis supports the feasibility of the universal MAP estimation approach.

Abstract

We study the compressed sensing (CS) signal estimation problem where an input signal is measured via a linear matrix multiplication under additive noise. While this setup usually assumes sparsity or compressibility in the input signal during recovery, the signal structure that can be leveraged is often not known a priori. In this paper, we consider universal CS recovery, where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself. Inspired by Kolmogorov complexity and minimum description length, we focus on a maximum a posteriori (MAP) estimation framework that leverages universal priors to match the complexity of the source. Our framework can also be applied to general linear inverse problems where more measurements than in CS might be needed. We provide theoretical results that support the algorithmic feasibility of universal MAP estimation using a Markov chain Monte Carlo implementation, which is computationally challenging. We incorporate some techniques to accelerate the algorithm while providing comparable and in many cases better reconstruction quality than existing algorithms. Experimental results show the promise of universality in CS, particularly for low-complexity sources that do not exhibit standard sparsity or compressibility.