Compression-Based Compressed Sensing

TL;DR

This paper explores the use of rate-distortion codes for compressed sensing of structured stochastic processes, demonstrating the optimality of the CSP algorithm in recovering signals from linear measurements in the low distortion regime.

Contribution

It extends the compressible signal pursuit (CSP) algorithm to stochastic processes and proves its reliability and optimality under certain conditions, linking rate-distortion dimension to information dimension.

Findings

CSP reliably recovers stationary process signals from linear measurements.

In low distortion regimes, the number of measurements needed is slightly more than the rate-distortion dimension times the signal length.

Universal recovery algorithms are developed using variable-length fixed-distortion compression codes.

Abstract

Modern compression algorithms exploit complex structures that are present in signals to describe them very efficiently. On the other hand, the field of compressed sensing is built upon the observation that "structured" signals can be recovered from their under-determined set of linear projections. Currently, there is a large gap between the complexity of the structures studied in the area of compressed sensing and those employed by the state-of-the-art compression codes. Recent results in the literature on deterministic signals aim at bridging this gap through devising compressed sensing decoders that employ compression codes. This paper focuses on structured stochastic processes and studies the application of rate-distortion codes to compressed sensing of such signals. The performance of the formerly-proposed compressible signal pursuit (CSP) algorithm is studied in this stochastic setting. It is proved that in the very low distortion regime, as the blocklength grows to infinity, the CSP algorithm reliably and robustly recovers $n$ instances of a stationary process from random linear projections as long as their count is slightly more than $n$ times the rate-distortion dimension (RDD) of the source. It is also shown that under some regularity conditions, the RDD of a stationary process is equal to its information dimension (ID). This connection establishes the optimality of the CSP algorithm at least for memoryless stationary sources, for which the fundamental limits are known. Finally, it is shown that the CSP algorithm combined by a family of universal variable-length fixed-distortion compression codes yields a family of universal compressed sensing recovery algorithms.