{"ID":5439478,"CreatedAt":"2026-07-01T01:17:58.482524686Z","UpdatedAt":"2026-07-02T18:49:48.32244458Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.30831","arxiv_id":"2606.30831","title":"Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices","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.","short_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 t...","url_abs":"https://arxiv.org/abs/2606.30831","url_pdf":"https://arxiv.org/pdf/2606.30831v1","authors":"[\"Mufan Li\",\"Jaume de Dios Pont\",\"Mihai Nica\",\"Daniel M. Roy\"]","published":"2026-06-29T19:01:43Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"stat.ML\"]","methods":"[]","has_code":false}