Noiseless compressed sensing refers to the problem of recovering a (high-dimensional) signal from its under-determined linear measurements. For compressed sensing to be feasible, the signal needs to be structured. While the main focus of the field has been on simple structures such as sparsity, there has been a growing interest in moving beyond sparsity and having a comprehensive compressed sensing framework that covers general structures. Two recent approaches that aim at developing such a framework from different perspectives are i) Quantized maximum a posteriori (Q-MAP), a Bayesian method that assumes full knowledge of the source distribution, and ii) Lagrangian minimum entropy pursuit (L-MEP), a universal recovery method that requires no prior knowledge about the distribution of the source. In this paper, by establishing theoretical connections between L-MEP and Q-MAP, it is shown how the two methods are complementary to each other and lead to a theoretically-founded learning-based recovery method that applies to sources with general structures. Unlike a Bayesian or a universal method, a learning-based method is able to extract the source structure from training data. The effect of error in estimating the source structure on the performance of the learning-based compressed sensing recovery method is characterized.

中文翻译：

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走向有理论基础的基于学习的压缩感知

无噪声压缩感知是指从不确定的线性测量中恢复（高维）信号的问题。为了使压缩感知可行，需要对信号进行结构化。虽然该领域的主要关注点是简单的结构，例如稀疏性，但人们对超越稀疏性并拥有涵盖一般结构的综合压缩感知框架越来越感兴趣。最近两种旨在从不同角度开发此类框架的方法是 i) 量化最大后验 (Q-MAP)，一种假设完全了解源分布的贝叶斯方法，以及 ii) 拉格朗日最小熵追踪 (L-MEP) ，一种通用的恢复方法，不需要有关源分布的先验知识。在本文中，通过在 L-MEP 和 Q-MAP 之间建立理论联系，展示了这两种方法如何相互补充，并导致理论基础的基于学习的恢复方法，适用于具有一般结构的源。与贝叶斯或通用方法不同，基于学习的方法能够从训练数据中提取源结构。描述了估计源结构中的错误对基于学习的压缩感知恢复方法性能的影响。