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arXiv · 1607.02335

The Mutual Information in Random Linear Estimation

Jean Barbier·Mohamad Dia·Nicolas Macris·Florent Krzakala
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Abstract

We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections, a problem relevant in compressed sensing, sparse superposition codes or code division multiple access just to cite few. There has been a number of works considering the mutual information for this problem using the heuristic replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical cases, we present a converse lower bound via a method that uses spatial coupling, state evolution analysis and the I-MMSE theorem. This yields, in particular, a single letter formula for the mutual information and the minimal-mean-square error for random Gaussian linear estimation of all discrete bounded signals.

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Jean Barbier, Mohamad Dia, Nicolas Macris, Florent Krzakala. 2016-07-08. The Mutual Information in Random Linear Estimation. https://doi.org/10.1109/allerton.2016.7852290

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