Projection divergence in the reproducing kernel Hilbert space: Asymptotic normality, block-wise and slicing estimation, and computational efficiency

We introduce projection divergence in the reproducing kernel Hilbert space to test for statistical independence and measure the degree of nonlinear dependence. We suggest a slicing procedure to estimate the kernel projection divergence, which divides a random sample of size $n$ into $H$ slices, each of size $c$. The entire procedure has the complexity of $O\left(n^2\right)$, which is prohibitive if $n$ is extremely large. To alleviate computational complexity, we implement this slicing procedure together with a block-wise estimation, which divides the whole sample into $B$ blocks, each of size $d$. This block-wise and slicing estimation has the complexity of $O\left\{n(c+d+logn)\right\}$, which reduces the computational complexity substantially if $c$ and $d$ are relatively small. The resultant estimation is asymptotically normal and has the convergence rate of ${\{n\left(cd\right)/(c+d)\}}^{-1/2}$. More importantly, this block-wise implementation has the same asymptotic properties as the naive slicing estimation, if $c$ is relatively small, indicating that the block-wise implementation does not result in power loss in independence tests. We demonstrate the computational efficiencies and theoretical properties of this block-wise and slicing estimation through simulations and an application to psychological datasets.