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Dynamical Functional Theory for Compressed Sensing
Citation key CaOpWiFle17
Author Burak Çakmak and Manfred Opper and Ole Winther and Bernard H. Fleury
Pages 2143-2147
Year 2017
DOI 10.1109/ISIT.2017.8006908
Journal 2017 IEEE International Symposium on Information Theory (ISIT)
Publisher IEEE Press
Abstract We introduce a theoretical approach for designing generalizations of the approximate message passing (AMP) algorithm for compressed sensing which are valid for large observation matrices that are drawn from an invariant random matrix ensemble. By design, the fixed points of the algorithm obey the Thouless-Anderson-Palmer (TAP) equations corresponding to the ensemble. Using a dynamical functional approach we are able to derive an effective stochastic process for the marginal statistics of a single component of the dynamics. This allows us to design memory terms in the algorithm in such a way that the resulting fields become Gaussian random variables allowing for an explicit analysis. The asymptotic statistics of these fields are consistent with the replica ansatz of the compressed sensing problem.
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