Fundamental limits of Non-Linear Low-Rank Matrix Estimation

arXiv:2403.04234v1 Announce Type: cross
Abstract: We consider the task of estimating a low-rank matrix from non-linear and noisy observations. We prove a strong universality result showing that Bayes-optimal performances are characterized by an equivalent Gaussian model with an effective prior, whose parameters are entirely determined by an expansion of the non-linear function. In particular, we show that to reconstruct the signal accurately, one requires a signal-to-noise ratio growing as $N^{\frac 12 (1-1/k_F)}$, where $k_F$ is the first non-zero Fisher information …

abstract arxiv bayes cs.lg expansion function linear low matrix non-linear parameters performances prior prove stat.ml type