Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices

• URL: http://arxiv.org/abs/2606.30831v1
• Date: Mon, 29 Jun 2026 19:01:43 GMT
• Title: Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices
• Abstract summary: We study the squared singular value spectrum of a product of non-square random matrices.<n>We first take a proportional depth-width $d,n$ limit with the number of data points $m$ held fixed, and show that the resulting covariance eigenvalue process satisfies a geometric version of Dyson Brownian motion.<n>We also use the limiting spectral law to predict a toy random-feature regression risk, finding close agreement with a finite-dimensional simulation.
• Score: 11.32927490081994
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the squared singular value spectrum of a product of non-square random matrices, a setting that also corresponds to the feature covariance eigenvalues of a deep linear neural network at initialization. We first take a proportional depth-width $d,n$ limit with the number of data points $m$ held fixed, and show that the resulting covariance eigenvalue process satisfies a geometric version of Dyson Brownian motion. We then take a second, sequential mean-field limit corresponding to the scaling $dm/n\to\barτ$, and show that the limiting $T$-transform of the spectrum solves a Burgers equation. In the identity-start case this equation yields the free log-normal law, and the general limit is obtained by free multiplicative convolution with the free log-normal. We further obtain the free log-normal support formula, a fixed-point iteration for numerical evaluation, and a formal small-time Marchenko--Pastur approximation. We also use the limiting spectral law to predict a toy random-feature regression risk, finding close agreement with a finite-dimensional simulation.

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