  Factorization of matrices where the rank of the two factors diverges linearly
with their sizes has many applications in diverse areas such as unsupervised
representation learning, dictionary learning or sparse coding. We consider a
setting where the two factors are generated from known component-wise
independent prior distributions, and the statistician observes a (possibly
noisy) component-wise function of their matrix product. In the limit where the
dimensions of the matrices tend to infinity, but their ratios remain fixed, we
expect to be able to derive closed form expressions for the optimal mean
squared error on the estimation of the two factors. However, this remains a
very involved mathematical and algorithmic problem. A related, but simpler,
problem is extensive-rank matrix denoising, where one aims to reconstruct a
matrix with extensive but usually small rank from noisy measurements. In this
paper, we approach both these problems using high-temperature expansions at
fixed order parameters. This allows to clarify how previous attempts at solving
these problems failed at finding an asymptotically exact solution. We provide a
systematic way to derive the corrections to these existing approximations,
taking into account the structure of correlations particular to the problem.
Finally, we illustrate our approach in detail on the case of extensive-rank
matrix denoising. We compare our results with known optimal
rotationally-invariant estimators, and show how exact asymptotic calculations
of the minimal error can be performed using extensive-rank matrix integrals.
