Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Gaussian Matrix Product
arXiv stat.ML1mo4 min read
arXiv:2606.30831v2 Announce Type: replace-cross Abstract: We study the squared singular value spectrum of a non-square product of independent real Gaussian matrices, equivalently the feature covariance spectrum of a deep linear neural network at initialization. Starting from the fixed-$m$ covariance diffusion previously obtained in the proportional depth-width limit, we record an equivalent matrix realization, describe its affine invariance, and derive the interacting diffusion satisfied by its eigenvalues. We then take a second limit, sending $m\to\infty$ on the accelerated spectral clock $\t
