Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices

TL;DR

This paper advances compressed sensing by developing probabilistic reconstruction algorithms, analyzing phase diagrams, and designing optimal measurement matrices to approach theoretical limits of data acquisition efficiency.

Contribution

It introduces a comprehensive probabilistic framework, new algorithms, and optimal measurement matrices for near-theoretical-limit compressed sensing performance.

Findings

Probabilistic approach achieves near-optimal reconstruction.

New seeding matrices improve measurement efficiency.

Phase diagrams characterize performance under noise and different signals.

Abstract

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.