Gradient flow on extensive-rank positive semi-definite matrix denoising
This work addresses matrix denoising challenges in high-dimensional data analysis, presenting incremental theoretical insights.
The paper tackles the problem of analyzing gradient flow for positive semi-definite matrix denoising in extensive-rank and high-dimensional settings, using random matrix theory to derive fixed point equations that track error evolution and uncovering continuous phase transitions.
In this work, we present a new approach to analyze the gradient flow for a positive semi-definite matrix denoising problem in an extensive-rank and high-dimensional regime. We use recent linear pencil techniques of random matrix theory to derive fixed point equations which track the complete time evolution of the matrix-mean-square-error of the problem. The predictions of the resulting fixed point equations are validated by numerical experiments. In this short note we briefly illustrate a few predictions of our formalism by way of examples, and in particular we uncover continuous phase transitions in the extensive-rank and high-dimensional regime, which connect to the classical phase transitions of the low-rank problem in the appropriate limit. The formalism has much wider applicability than shown in this communication.