The spiked matrix mannequin with generative priors
Authors: Benjamin Aubin, Bruno Loureiro, Antoine Maillard, Florent Krzakala, Lenka Zdeborová
Summary: Utilizing a low-dimensional parametrization of alerts is a generic and highly effective option to improve efficiency in sign processing and statistical inference. A highly regarded and extensively explored sort of dimensionality discount is sparsity; one other sort is generative modelling of sign distributions. Generative fashions primarily based on neural networks, resembling GANs or variational auto-encoders, are significantly performant and are gaining on applicability. On this paper we examine spiked matrix fashions, the place a low-rank matrix is noticed by way of a loud channel. This downside with sparse construction of the spikes has attracted broad consideration prior to now literature. Right here, we exchange the sparsity assumption by generative modelling, and examine the results on statistical and algorithmic properties. We analyze the Bayes-optimal efficiency underneath particular generative fashions for the spike. In distinction with the sparsity assumption, we don’t observe areas of parameters the place statistical efficiency is superior to the perfect recognized algorithmic efficiency. We present that within the analyzed instances the approximate message passing algorithm is ready to attain optimum efficiency. We additionally design enhanced spectral algorithms and analyze their efficiency and thresholds utilizing random matrix idea, displaying their superiority to the classical principal element evaluation. We complement our theoretical outcomes by illustrating the efficiency of the spectral algorithms when the spikes come from actual datasets
