The spectrum of random kernel matrices: universality results for rough and varying kernels

2012/06/17 by Yen Do, Van Vu, Do, Yen +1 · 6 citations
Mathematics · Computer Science · #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Bayesian Methods and Mixture Models

paper · pdf · doi:10.48550/arxiv.1206.3763

Abstract

We consider random matrices whose entries are f() or f(||Xi-Xj||²) for iid vectors Xi in R^p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui [17] showed that the spectral distributions of these matrices behave as if f is linear in the Marchenko--Pastur limit. When Xi's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng and Singer [13]. Two results are shown in this paper: first it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and secondly it is shown that the Gaussian assumptions in Cheng--Singer's result can be removed, answering a question posed in [13] about the universality of the limiting spectral distribution.

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