Universality in polytope phase transitions and iterative algorithms

Mohsen Bayati, Marc Lelarge, Andrea Montanari

Abstract: We consider a class of nonlinear mappings F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A, N</sub> in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> indexed by symmetric random matrices A ϵ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N×N</sup> with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are known as `approximate message passing' algorithms. We study the high-dimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal, i.e. it depends only on the first two moments of the entries of A. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves a conjecture by David Donoho and Jared Tanner.