Optimal Kernel Combination for Test of Independence against Local Alternatives

TL;DR

This paper introduces a method to select the optimal kernel pair for independence testing, enhancing test power against local alternatives by leveraging the asymptotic null distribution.

Contribution

It proposes a novel kernel selection strategy based on local alternatives, improving the efficiency and power of HSIC-based independence tests.

Findings

Higher test power compared to existing kernel selection methods

Effective kernel pair selection using local alternatives

Improved test efficiency without dependence on specific alternatives

Abstract

Testing the independence between two random variables $x$ and $y$ is an important problem in statistics and machine learning, where the kernel-based tests of independence is focused to address the study of dependence recently. The advantage of the kernel framework rests on its flexibility in choice of kernel. The Hilbert-Schmidt Independence Criterion (HSIC) was shown to be equivalent to a class of tests, where the tests are based on different distance-induced kernel pairs. In this work, we propose to select the optimal kernel pair by considering local alternatives, and evaluate the efficiency using the quadratic time estimator of HSIC. The local alternative offers the advantage that the measure of efficiency do not depend on a particular alternative, and only requires the knowledge of the asymptotic null distribution of the test. We show in our experiments that the proposed strategy results in higher power than other existing kernel selection approaches.