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Compression-Based Compressed Sensing

机译：基于压缩的压缩感知

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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 instances of a stationary process from its random linear measurements as long as is slightly more than 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.
机译：现代压缩代码利用信号的复杂结构来对其进行高效编码。另一方面，压缩传感算法从不确定的线性测量集中恢复“结构化”信号。当前，在压缩感测领域中使用的结构类型与最新压缩代码所使用的结构类型之间存在明显的差距。关于确定性信号的文献中的最新结果旨在通过设计采用压缩代码的压缩感测解码器来弥合这一差距。本文着重于结构化随机过程，并研究有损压缩码在此类信号的压缩感知中的应用。在这种随机环境下，研究了以前提出的可压缩信号追踪（CSP）优化的性能。事实证明，在低失真状态下，随着块长增长到无穷大，CSP优化能够可靠，可靠地从其随机线性测量中恢复平稳过程的实例，只要它稍大于速率失真维数（RDD）的几倍即可。 ）的来源。还表明，在某些规则条件下，平稳过程的RDD等于其信息维。这种连接至少为已知的基本限制的无记忆固定源建立了CSP的最优性。最后，证明了通过一系列通用可变长度固定失真压缩码组合的CSP产生了一系列通用压缩感测恢复算法。

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