The spiked matrix model with generative priors
Authors: Benjamin Aubin, Bruno Loureiro, Antoine Maillard, Florent Krzakala, Lenka Zdeborová
Abstract: Using a low-dimensional parametrization of alerts is a generic and extremely efficient possibility to enhance effectivity in signal processing and statistical inference. A extremely regarded and extensively explored kind of dimensionality low cost is sparsity; one different type is generative modelling of signal distributions. Generative fashions based on neural networks, resembling GANs or variational auto-encoders, are considerably performant and are gaining on applicability. On this paper we study spiked matrix fashions, the place a low-rank matrix is observed by the use of a loud channel. This draw back with sparse development of the spikes has attracted broad consideration before now literature. Proper right here, we alternate the sparsity assumption by generative modelling, and study the outcomes on statistical and algorithmic properties. We analyze the Bayes-optimal effectivity beneath explicit generative fashions for the spike. In distinction with the sparsity assumption, we do not observe areas of parameters the place statistical effectivity is superior to the proper acknowledged algorithmic effectivity. We current that inside the analyzed situations the approximate message passing algorithm is able to attain optimum effectivity. We moreover design enhanced spectral algorithms and analyze their effectivity and thresholds using random matrix concept, displaying their superiority to the classical principal aspect analysis. We complement our theoretical outcomes by illustrating the effectivity of the spectral algorithms when the spikes come from precise datasets
