Efficient Aggregated Kernel Tests using Incomplete $U$-statistics

• URL: http://arxiv.org/abs/2206.09194v1
• Date: Sat, 18 Jun 2022 12:30:06 GMT
• Title: Efficient Aggregated Kernel Tests using Incomplete $U$-statistics
• Authors: Antonin Schrab and Ilmun Kim and Benjamin Guedj and Arthur Gretton
• Abstract summary: Three proposed tests aggregate over several kernel bandwidths to detect departures from the null on various scales. We show that our proposed linear-time aggregated tests obtain higher power than current state-of-the-art linear-time kernel tests.
• Score: 22.251118308736327
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We propose a series of computationally efficient, nonparametric tests for the two-sample, independence and goodness-of-fit problems, using the Maximum Mean Discrepancy (MMD), Hilbert Schmidt Independence Criterion (HSIC), and Kernel Stein Discrepancy (KSD), respectively. Our test statistics are incomplete $U$-statistics, with a computational cost that interpolates between linear time in the number of samples, and quadratic time, as associated with classical $U$-statistic tests. The three proposed tests aggregate over several kernel bandwidths to detect departures from the null on various scales: we call the resulting tests MMDAggInc, HSICAggInc and KSDAggInc. For the test thresholds, we derive a quantile bound for wild bootstrapped incomplete $U$- statistics, which is of independent interest. We derive uniform separation rates for MMDAggInc and HSICAggInc, and quantify exactly the trade-off between computational efficiency and the attainable rates: this result is novel for tests based on incomplete $U$-statistics, to our knowledge. We further show that in the quadratic-time case, the wild bootstrap incurs no penalty to test power over more widespread permutation-based approaches, since both attain the same minimax optimal rates (which in turn match the rates that use oracle quantiles). We support our claims with numerical experiments on the trade-off between computational efficiency and test power. In the three testing frameworks, we observe that our proposed linear-time aggregated tests obtain higher power than current state-of-the-art linear-time kernel tests.

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