  Approximate Message Passing (AMP) algorithms provide a valuable tool for
studying mean-field approximations and dynamics in a variety of applications.
Although these algorithms are often first derived for matrices having
independent Gaussian entries or satisfying rotational invariance in law, their
state evolution characterizations are expected to hold over larger universality
classes of random matrix ensembles.
  We develop several new results on AMP universality. For AMP algorithms
tailored to independent Gaussian entries, we show that their state evolutions
hold over broadly defined generalized Wigner and white noise ensembles,
including matrices with heavy-tailed entries and heterogeneous entrywise
variances that may arise in data applications. For AMP algorithms tailored to
rotational invariance in law, we show that their state evolutions hold over
delocalized sign-and-permutation-invariant matrix ensembles that have a limit
distribution over the diagonal, including sensing matrices composed of
subsampled Hadamard or Fourier transforms and diagonal operators.
  We establish these results via a simplified moment-method proof, reducing AMP
universality to the study of products of random matrices and diagonal tensors
along a tensor network. As a by-product of our analyses, we show that the
aforementioned matrix ensembles satisfy a notion of asymptotic freeness with
respect to such tensor networks, which parallels usual definitions of freeness
for traces of matrix products.
