Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime

• URL: http://arxiv.org/abs/2310.18280v1
• Date: Fri, 27 Oct 2023 17:15:55 GMT
• Title: Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime
• Authors: Sofiia Dubova, Yue M. Lu, Benjamin McKenna, Horng-Tzer Yau
• Abstract summary: In this paper, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments. In the case of non-integer $ell$, the Marvcenko-Pastur term disappears.
• Score: 12.221087476416056
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a nonlinear function $f$ applied entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \mathbb{R}^{d \times N}$ is random and normalized in such a way that $f$ typically has order-one arguments. We work in the polynomial regime, where $N \asymp d^\ell$ for some $\ell > 0$, not just the linear regime where $\ell = 1$. 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. In this paper, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments. In the case of non-integer $\ell$, the Mar\v{c}enko-Pastur term disappears (its weight in the free convolution vanishes), and the spectrum is just semicircular.

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