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Abstract:
The problem of sparse representations in redundant systems may conditionally be considered as "compression". At the same time, it is essentially equivalent to many other real world applied problems. We are going to discuss its equivalency to the problems of compressed sensing, error correcting codes, and in-painting.
The mathematical equivalency of those problems brings the opportunity for mutual enrichment of the theoretical and numerical methods for their solutions. We discuss three types of numerical algorithms (Linear Programming, Orthogonal Greedy Algorithm, and Dynamic Programming) and their perspectives for sparse representations.
We show how the algorithms developed for sparse representations can be used in "anti-sparse" settings, e.g., for the problem of the consistent restoration of data from quantized redundant transforms.
Roughly speaking, in the framework of Shannon's Information Theory, the mentioned problems can be considered either as related to compression or to reliable transmission through noisy channels. However, Shannon's theory also contains Cryptography as its part. In the end of the talk, we give our view how the ideas arising in sparse representations may be used in cryptographic codes with an open key.
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