On the spectral property of kernel-based sensor fusion algorithms of high dimensional data

TL;DR

This paper investigates the spectral properties of kernel-based sensor fusion algorithms, NCCA and AD, in high-dimensional settings using random matrix theory and free probability, revealing eigenvalue behaviors under independence assumptions.

Contribution

It provides a theoretical analysis of the eigenvalue convergence of NCCA and AD algorithms in high-dimensional regimes, especially for Gaussian data, using advanced mathematical tools.

Findings

Eigenvalues of kernel matrices converge to deterministic limits for Gaussian data.

The convergence rate of kernel affinity matrix eigenvalues is established.

Spectral properties depend on data independence and kernel smoothness.

Abstract

We apply local laws of random matrices and free probability theory to study the spectral properties of two kernel-based sensor fusion algorithms, nonparametric canonical correlation analysis (NCCA) and alternating diffusion (AD), for two simultaneously recorded high dimensional datasets under the null hypothesis. The matrix of interest is the product of the kernel matrices associated with the databsets, which may not be diagonalizable in general. We prove that in the regime where dimensions of both random vectors are comparable to the sample size, if NCCA and AD are conducted using a smooth kernel function, then the first few nontrivial eigenvalues will converge to real deterministic values provided the datasets are independent Gaussian random vectors. Toward the claimed result, we also provide a convergence rate of eigenvalues of a kernel affinity matrix.