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…

• EC
  European Commission

  Awards: 884584, 101039794
• NS
  Natural Sciences and Engineering Research Council of Canada

  Awards: RGPIN-2020, RGPIN-2020-04597