Gradient flow on extensive-rank positive semi-definite matrix denoising

Antoine BodinNicolas Macris

Mar 2023

摘要

In this work, we present a new approach to analyze the gradient flow for apositive semi-definite matrix denoising problem in an extensive-rank andhigh-dimensional regime. We use recent linear pencil techniques of randommatrix theory to derive fixed point equations which track the complete timeevolution of the matrix-mean-square-error of the problem. The predictions ofthe resulting fixed point equations are validated by numerical experiments. Inthis short note we briefly illustrate a few predictions of our formalism by wayof examples, and in particular we uncover continuous phase transitions in theextensive-rank and high-dimensional regime, which connect to the classicalphase transitions of the low-rank problem in the appropriate limit. Theformalism has much wider applicability than shown in this communication.

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