  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.
