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April 2015 Universality in polytope phase transitions and message passing algorithms
Mohsen Bayati, Marc Lelarge, Andrea Montanari
Ann. Appl. Probab. 25(2): 753-822 (April 2015). DOI: 10.1214/14-AAP1010


We consider a class of nonlinear mappings $\mathsf{F}_{A,N}$ in $\mathbb{R}^{N}$ indexed by symmetric random matrices $A\in\mathbb{R}^{N\times N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333–366]. Within information theory, they are known as “approximate message passing” algorithms.

We study the high-dimensional (large $N$) behavior of the iterates of $\mathsf{F}$ for polynomial functions $\mathsf{F}$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.


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Mohsen Bayati. Marc Lelarge. Andrea Montanari. "Universality in polytope phase transitions and message passing algorithms." Ann. Appl. Probab. 25 (2) 753 - 822, April 2015.


Published: April 2015
First available in Project Euclid: 19 February 2015

zbMATH: 1322.60207
MathSciNet: MR3313755
Digital Object Identifier: 10.1214/14-AAP1010

Primary: 60F05
Secondary: 68W40

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 2015
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