A multiscale cavity method for sublinear-rank symmetric matrix factorization
<|reference_start|>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 $M$ of the signal matrix to infer scales with its size $N$ as $M = o(N^{1/10})$. Allowing for a $N$-dependent rank offers new challenges and requires new methods. Working in the Bayesian-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 Gaussian vector 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.<|reference_end|>
@article{barbier2024a,
    title={A multiscale cavity method for sublinear-rank symmetric matrix
  factorization},
    author={Jean Barbier, Justin Ko and Anas A. Rahman},
    journal={arXiv preprint arXiv:2403.07189},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.07189},
    primaryClass={cs.IT cond-mat.dis-nn math-ph math.IT math.MP math.ST stat.TH}
}