Compression-Based Compressed Sensing

Abstract

Modern compression codes exploit signals’ complex structures to encode them very efficiently. On the other hand, compressed sensing algorithms recover “structured” signals from their under-determined set of linear measurements. Currently, there is a noticeable gap between the types of structures used in the area of compressed sensing and those employed by 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 application of lossy compression codes to compressed sensing of such signals. The performance of the formerly proposed compressible signal pursuit (CSP) optimization is studied in this stochastic setting. It is proved that in the low-distortion regime, as the blocklength grows to infinity, the CSP optimization reliably and robustly recovers $n$ instances of a stationary process from its random linear measurements as long as $n$ 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. This connection establishes the optimality of CSP at least for memoryless stationary sources, which have known fundamental limits. Finally, it is shown that CSP combined by a family of universal variable-length fixed-distortion compression codes yields a family of universal compressed sensing recovery algorithms.