局部工作全局思考-第一部分：卷积稀疏编码的理论保证
Working Locally Thinking Globally - Part I: Theoretical Guarantees for Convolutional Sparse Coding
摘要：
在过去的十年中，著名的稀疏表示模型在各种信号处理任务中取得了显着的成果。然而，尽管其最初的目的是作为整个信号的全局先验，但由于使用学习词典时所需的计算约束，它通常用于建模低维补丁。最近已经提出了一种解决这个问题的方法，采用卷积稀疏表示模型。这种方法假设全局字典是带状循环矩阵的级联。虽然一些作品已经提出了算法解决方案的全球追求问题，在这个新的模式下，很少真正有效的保证是已知的成功，这样的方法。在这两部分工作的第一部分中，我们讨论了稀疏卷积模型的理论方面，为解决方案的唯一性和追踪算法的成功性的相应问题提供了第一个有意义的答案。为此，我们将数学量，如$\ell_0$范数，相互相干性和火花，推广到卷积设置中的对应物，其本质上捕获全局模型的局部测量。在一个配套文件中，我们扩展到一个嘈杂的制度，解决的稳定性的稀疏的解决方案和追求算法的分析，并通过简单的本地处理解决全球追求问题的实际方法。
英文摘要：
The celebrated sparse representation model has led to remarkable results in various signal processing tasks in the last decade. However, despite its initial purpose of serving as a global prior for entire signals, it has been commonly used for modeling low dimensional patches due to the computational constraints it entails when deployed with learned dictionaries. A way around this problem has been proposed recently, adopting a convolutional sparse representation model. This approach assumes that the global dictionary is a concatenation of banded Circulant matrices. Although several works have presented algorithmic solutions to the global pursuit problem under this new model, very few truly-effective guarantees are known for the success of such methods. In the first of this two-part work, we address the theoretical aspects of the sparse convolutional model, providing the first meaningful answers to corresponding questions of uniqueness of solutions and success of pursuit algorithms. To this end, we generalize mathematical quantities, such as the $\ell_0$ norm, the mutual coherence and the Spark, to their counterparts in the convolutional setting, which intrinsically capture local measures of the global model. In a companion paper, we extend the analysis to a noisy regime, addressing the stability of the sparsest solutions and pursuit algorithms, and demonstrate practical approaches for solving the global pursuit problem via simple local processing.
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