The Spiked Matrix Model With Generative Priors

We investigate the statistical and algorithmic properties of random neural-network generative priors in a simple inference problem: spiked-matrix estimation. establish rigorous expression for performance Bayes-optimal estimator high-dimensional regime, identify threshold weak-recovery spike. Next, we derive message-passing algorithm taking into account latent structure spike, show that its is asymptotically optimal natural choices network architecture. The absence an gap this case stark contrast to known results sparse spikes, another popular prior modelling low-dimensional signals, which no achieve threshold. Finally, linearising our message passing yields spectral method also achieving reconstruction. conclude with experiment on real data set showing bespoke outperforms vanilla PCA.