We consider a class of nonlinear mappings $mathsf{F}_{A,N}$ in $mathbb{R}^N$ indexed by symmetric random matrices $Ainmathbb{R}^{Ntimes N}$ 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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