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Factorization of matrices where the rank of the two factors diverges linearlywith their sizes has many applications in diverse areas such as unsupervisedrepresentation learning, dictionary learning or sparse coding. We consider asetting where the two factors are generated from known component-wiseindependent prior distributions, and the statistician observes a (possiblynoisy) component-wise function of their matrix product. In the limit where thedimensions of the matrices tend to infinity, but their ratios remain fixed, weexpect to be able to derive closed form expressions for the optimal meansquared error on the estimation of the two factors. However, this remains avery involved mathematical and algorithmic problem. A related, but simpler,problem is extensive-rank matrix denoising, where one aims to reconstruct amatrix with extensive but usually small rank from noisy measurements. In thispaper, we approach both these problems using high-temperature expansions atfixed order parameters. This allows to clarify how previous attempts at solvingthese problems failed at finding an asymptotically exact solution. We provide asystematic 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-rankmatrix denoising. We compare our results with known optimalrotationally-invariant estimators, and show how exact asymptotic calculationsof the minimal error can be performed using extensive-rank matrix integrals.

Perturbative construction of mean-field equations in extensive-rank matrix factorization and denoising