  We consider a statistical model for symmetric matrix factorization with
additive Gaussian noise in the high-dimensional regime where the rank $M$ of
the signal matrix to infer scales with its size $N$ as $M = o(N^{1/10})$.
Allowing for a $N$-dependent rank offers new challenges and requires new
methods. Working in the Bayesian-optimal setting, we show that whenever the
signal has i.i.d. entries the limiting mutual information between signal and
data is given by a variational formula involving a rank-one replica symmetric
potential. In other words, from the information-theoretic perspective, the case
of a (slowly) growing rank is the same as when $M = 1$ (namely, the standard
spiked Wigner model). The proof is primarily based on a novel multiscale cavity
method allowing for growing rank along with some information-theoretic
identities on worst noise for the Gaussian vector channel. We believe that the
cavity method developed here will play a role in the analysis of a broader
class of inference and spin models where the degrees of freedom are large
arrays instead of vectors.
