In double asymptotic limits, the squared singular value process of non-square matrix products obeys geometric Dyson Brownian motion whose T-transform solves a Burgers equation, producing the free log-normal law via free multiplicative convolution.
arXiv preprint arXiv:2408.01062 , year=
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Extends high-dimensional KRR to product kernels, proving convergence rates that recover minimax optimality for source condition s ≤ 1, saturation for s > 1, and multiple-descent phenomena with respect to sample size n.
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Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices
In double asymptotic limits, the squared singular value process of non-square matrix products obeys geometric Dyson Brownian motion whose T-transform solves a Burgers equation, producing the free log-normal law via free multiplicative convolution.