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Universal Compressed Sensing for Almost Lossless Recovery

机译：通用压缩传感，几乎无损恢复

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In this paper, the problem of developing universal algorithms for noiseless compressed sensing of stochastic processes is studied. First, Rényi's notion of information dimension (ID) is generalized to continuous-valued discrete-time stationary processes. This provides a measure of complexity for such processes and is connected to the rate of measurement (sampling rate) required for their accurate recovery. Then, based on Occam's razor, a minimum entropy pursuit (MEP) optimization approach for universal compressed sensing is proposed. It is proven that, for any stationary process satisfying certain mixing conditions, if the sampling rate is larger than the ID of the source process, MEP optimization can reliably recover the source vector almost losslessly, without having any prior information about its distribution. Then, a Lagrangian-type relaxation of MEP optimization problem, referred to as Lagrangian-MEP, is studied. It is shown that Lagrangian-MEP is identical to an implementable algorithm proposed by Baron and coauthors, and for the right choice of parameters, has the same asymptotic performance as MEP optimization. Finally, it is proven that Lagrangian-MEP is robust to small measurement noise.
机译：本文研究了开发用于随机过程的无噪声压缩感知的通用算法的问题。首先，雷尼（Rényi）的信息维（ID）概念被推广到连续值离散时间平稳过程。这为此类过程提供了一种复杂程度的度量，并与它们的准确恢复所需的度量速率（采样速率）相关。然后，基于Occam的剃刀，提出了一种用于通用压缩感知的最小熵追踪（MEP）优化方法。事实证明，对于任何满足某些混合条件的平稳过程，如果采样率大于源过程的ID，则MEP优化可以可靠地几乎无损地恢复源向量，而无需任何有关其分布的先验信息。然后，研究了拉格朗日型MEP优化问题的松弛，称为拉格朗日MEP。结果表明，Lagrangian-MEP与Baron及其合作者提出的可实现算法相同，并且对于正确的参数选择，其渐近性能与MEP优化相同。最终，证明了拉格朗日MEP对小测量噪声具有鲁棒性。

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