  We consider a statistical model for matrix factorization in a regime where
the rank of the two hidden matrix factors grows linearly with their dimension
and their product is corrupted by additive noise. Despite various approaches,
statistical and algorithmic limits of such problems have remained elusive. We
study a Bayesian setting with the assumptions that (a) one of the matrix
factors is symmetric, (b) both factors as well as the additive noise have
rotational invariant priors, (c) the priors are known to the statistician. We
derive analytical formulas for Rotation Invariant Estimators to reconstruct the
two matrix factors, and conjecture that these are optimal in the
large-dimension limit, in the sense that they minimize the average
mean-square-error. We provide numerical checks which confirm the optimality
conjecture when confronted to Oracle Estimators which are optimal by
definition, but involve the ground-truth. Our derivation relies on a
combination of tools, namely random matrix theory transforms, spherical
integral formulas, and the replica method from statistical mechanics.
