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.
Celli.Wide Neural Networks with General Weights: Convergence Rate and Explicit Depen- dence on the Hyper-Parameters
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Functionals of infinite-width random neural networks on the sphere exhibit phase transitions in fluctuations as depth grows, converging to a limiting Gaussian field functional, a Gaussian, or a Qth Wiener chaos distribution depending on covariance fixed points.
Proves total variation distance between finite neural network output laws and their order-(4m-1) Edgeworth approximations is O(n^{-m}) with matching lower bounds, under invertible covariance and polynomially bounded activations; extends to conditionally Gaussian sequences.
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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.
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Phase Transitions in the Fluctuations of Functionals of Random Neural Networks
Functionals of infinite-width random neural networks on the sphere exhibit phase transitions in fluctuations as depth grows, converging to a limiting Gaussian field functional, a Gaussian, or a Qth Wiener chaos distribution depending on covariance fixed points.