FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test

Abstract

The maximum mean discrepancy (MMD) is a recently proposed test statistic for the two-sample test. Its quadratic time complexity, however, greatly hampers its availability to large-scale applications. To accelerate the MMD calculation, in this study we propose an efficient method called FastMMD. The core idea of FastMMD is to equivalently transform the MMD with shift-invariant kernels into the amplitude expectation of a linear combination of sinusoid components based on Bochner's theorem and Fourier transform (Rahimi & Recht, 2007). Taking advantage of sampling the Fourier transform, FastMMD decreases the time complexity for MMD calculation from O(N(2)d) to O(LN d), where N and d are the size and dimension of the sample set, respectively. Here, L is the number of basis functions for approximating kernels that determines the approximation accuracy. For kernels that are spherically invariant, the computation can be further accelerated to O(LN log d) by using the Fastfood technique (Le, Sarlós, & Smola, 2013). The uniform convergence of our method has also been theoretically proved in both unbiased and biased estimates. We also provide a geometric explanation for our method, ensemble of circular discrepancy, which helps us understand the insight of MMD and we hope will lead to more extensive metrics for assessing the two-sample test task. Experimental results substantiate that the accuracy of FastMMD is similar to that of MMD and with faster computation and lower variance than existing MMD approximation methods.