  Approximate message passing algorithms proved to be extremely effective in
reconstructing sparse signals from a small number of incoherent linear
measurements. Extensive numerical experiments further showed that their
dynamics is accurately tracked by a simple one-dimensional iteration termed
state evolution. In this paper we provide the first rigorous foundation to
state evolution. We prove that indeed it holds asymptotically in the large
system limit for sensing matrices with independent and identically distributed
gaussian entries.
  While our focus is on message passing algorithms for compressed sensing, the
analysis extends beyond this setting, to a general class of algorithms on dense
graphs. In this context, state evolution plays the role that density evolution
has for sparse graphs.
  The proof technique is fundamentally different from the standard approach to
density evolution, in that it copes with large number of short loops in the
underlying factor graph. It relies instead on a conditioning technique recently
developed by Erwin Bolthausen in the context of spin glass theory.
