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We propose a class of nonparametric two-sample tests with a cost linear inthe sample size. Two tests are given, both based on an ensemble of distancesbetween analytic functions representing each of the distributions. The firsttest uses smoothed empirical characteristic functions to represent thedistributions, the second uses distribution embeddings in a reproducing kernelHilbert space. Analyticity implies that differences in the distributions may bedetected almost surely at a finite number of randomly chosenlocations/frequencies. The new tests are consistent against a larger class ofalternatives than the previous linear-time tests based on the (non-smoothed)empirical characteristic functions, while being much faster than the currentstate-of-the-art quadratic-time kernel-based or energy distance-based tests.Experiments on artificial benchmarks and on challenging real-world testingproblems demonstrate that our tests give a better power/time tradeoff thancompeting approaches, and in some cases, better outright power than even themost expensive quadratic-time tests. This performance advantage is retainedeven in high dimensions, and in cases where the difference in distributions isnot observable with low order statistics.

Fast Two-Sample Testing with Analytic Representations of Probability Measures