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Working Locally Thinking Globally - Part I: Theoretical Guarantees for Convolutional Sparse Coding by Papyan, Vardan; Sulam, Jeremias; Elad, Michael is a scholarly article available to read on EtoBox.

What is Working Locally Thinking Globally - Part I: Theoretical Guarantees for Convolutional Sparse Coding about?

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 me

Author

Papyan, Vardan; Sulam, Jeremias; Elad, Michael

Published

2016

Language

EN