A multiscale cavity method for sublinear-rank symmetric matrix factorization

We consider a statistical model for symmetric matrix factorization with additive Gaussian noise in the high-dimensional regime, where the rank of the signal matrix to infer M scales with its size N as M=\mathrm{o}(\sqrt{\ln N}) . Allowing for an N -dependent rank offers new challenges and requires new methods. Working in the Bayes-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 vector Gaussian 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.