  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.
