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

In this paper, 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 sequences of random vectors $\mathcal{X}:=\{\xb_i\}_{i=1}^n$ and $\mathcal{Y}:=\{\yb_i\}_{i=1}^n$ under the null hypothesis. The matrix of interest is a product of the kernel matrices associated with $\mathcal{X}$ and $\mathcal{Y}$, 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 $\mathcal{X}$ and $\mathcal{Y}$ are independent Gaussian random vectors. We propose an eigenvalue-ratio test based on the real parts of the eigenvalues of the product matrix to test if $\mathcal{X}$ and $\mathcal{Y}$ are independent and do not share common information. Simulation study verifies the usefulness of such statistic.