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The celebrated sparse representation model has led to remarkable results invarious signal processing tasks in the last decade. However, despite itsinitial purpose of serving as a global prior for entire signals, it has beencommonly used for modeling low dimensional patches due to the computationalconstraints it entails when deployed with learned dictionaries. A way aroundthis problem has been proposed recently, adopting a convolutional sparserepresentation model. This approach assumes that the global dictionary is aconcatenation of banded Circulant matrices. Although several works havepresented algorithmic solutions to the global pursuit problem under this newmodel, very few truly-effective guarantees are known for the success of suchmethods. In the first of this two-part work, we address the theoretical aspectsof the sparse convolutional model, providing the first meaningful answers tocorresponding questions of uniqueness of solutions and success of pursuitalgorithms. To this end, we generalize mathematical quantities, such as the$\ell_0$ norm, the mutual coherence and the Spark, to their counterparts in theconvolutional setting, which intrinsically capture local measures of the globalmodel. In a companion paper, we extend the analysis to a noisy regime,addressing the stability of the sparsest solutions and pursuit algorithms, anddemonstrate practical approaches for solving the global pursuit problem viasimple local processing.

Working Locally Thinking Globally - Part I: Theoretical Guarantees for Convolutional Sparse Coding