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Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime
Earlier work by various authors showed that, when the columns of $X$ are either uniform on the sphere or standard Gaussian vectors, and when $\ell$ is an integer (the linear regime $\ell = 1$ is particularly well-studied), the bulk eigenvalues of such matrices behave in a simple way: They are asymptotically given by the free convolution of the semicircular and Mar\v{c}enko-Pastur distributions, with relative weights given by expanding $f$ in the Hermite basis.
An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings
Our work reveals an equivalence principle: the spectrum of the random kernel matrix is asymptotically equivalent to that of a simpler matrix model, constructed as a linear combination of a (shifted) Wishart matrix and an independent matrix sampled from the Gaussian orthogonal ensemble.
Spectrum of Random $d$-regular Graphs Up to the Edge
Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq3$.
Probability Mathematical Physics Combinatorics Mathematical Physics 60B20, 05C80
Dynamics of Deep Neural Networks and Neural Tangent Hierarchy
However, it was observed in [5] that there is a performance gap between the kernel regression using the limiting NTK and the deep neural networks.
