Spectrum of large random inner-product kernel matrices generated from \(l_p\) ellipsoids. (English) Zbl 07529918
Summary: In this note, we study the \(n \times n\) random kernel matrix whose \((i,j)^{th}\) entry is of form \(f(\mathbf{x}_i^\prime \mathbf{x}_j) -u_n\delta_{ij}\sum_k f(\mathbf{x}_i^\prime \mathbf{x}_k)\) for some real function \(f\) and the \(x_i\)’s are i.i.d. random vectors generated from \(l_p\) ellipsoid or its surface in \(\mathbb{R}^N\). The limit of the empirical spectral distribution is derived in the regime where \(n\) and \(N\) grow proportionally to infinity. Here \(u_n\) is allowed to be any real number which includes the two most interesting cases \(u_n=0\) and \(u_n=1\). Besides, compared with the existing related work, our results require only minimal regularity assumptions on the kernel function \(f\).
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 60F99 | Limit theorems in probability theory |
| 62-XX | Statistics |
Keywords:
inner-product kernel matrices; empirical spectral distribution; Laplacian; high dimension; \(l_p\) ellipsoidsReferences:
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