B-tests: Low Variance Kernel Two-Sample Tests

By Wojciech Zaremba, Arthur Gretton, Matthew B. Blaschko. 2013

24 citations recorded by the source corpus.

A family of maximum mean discrepancy (MMD) kernel two-sample tests is introduced. Members of the test family are called Block-tests or B-tests, since the test statistic is an average over MMDs computed on subsets of the samples. The choice of block size allows control over the tradeoff between test power and computation time. In this respect, the $B$-test family combines favorable properties of previously proposed MMD two-sample tests: B-tests are more powerful than a linear time test where blocks are just pairs of samples, yet they are more computationally efficient than a quadratic time test where a single large block incorporating all the samples is used to compute a U-statistic. A further important advantage of the B-tests is their asymptotically Normal null distribution: this is by contrast with the U-statistic, which is degenerate under the null hypothesis, and for which estimates of the null distribution are computationally demanding. Recent results on kernel selection for hypothesis testing transfer seamlessly to the B-tests, yielding a means to optimize test power via kernel choice.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

A Kernelized Stein Discrepancy for Goodness-of-fit Tests (2016, methodology): “Because we can show that E x ∼ q [ u q ( x, x (cid:48) )] = 0 following the proof above for p = q , we must have c = 0 , and hence S ( p, q ) = E x ∼ p E x (cid:48) ∼ p [ u q ( x, x (cid:48) )] = c = 0 , which contradicts with p (cid:54) = q .”
FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test (2014, methodology): “Following previous work on kernel hypothesis testing (Gretton, Sriperumbudur et al., 2012; Zaremba et al., 2013), our synthetic distribution is designed as 5 × 5 grids of 2D gaussian blobs.”
Learning Deep Kernels for Non-Parametric Two-Sample Tests (2020, methodology): “Estimating the MMD with mini-batches corresponds to the block estimator approach of Zaremba et al. (2013).”
Demystifying MMD GANs (2018, methodology): “(12)Because the computation of the MMD estimator scales like O(n(2)d), we recommend using a relatively small n and averaging over several estimates; this is closely related to the block estimator of Zaremba et al. (2013). The FID estimator, for comparison, takes time O(nd(2) + d(3)), and is substant”