Fast Two-Sample Testing with Analytic Representations of Probability Measures
This work provides faster and more powerful statistical tests for comparing distributions, which is incremental but addresses a bottleneck in large-scale data analysis.
The authors tackled the problem of two-sample testing with a new class of nonparametric tests that achieve linear-time cost, offering better power/time tradeoffs than existing methods, including outperforming quadratic-time tests in some cases, as demonstrated on artificial and real-world benchmarks.
We propose a class of nonparametric two-sample tests with a cost linear in the sample size. Two tests are given, both based on an ensemble of distances between analytic functions representing each of the distributions. The first test uses smoothed empirical characteristic functions to represent the distributions, the second uses distribution embeddings in a reproducing kernel Hilbert space. Analyticity implies that differences in the distributions may be detected almost surely at a finite number of randomly chosen locations/frequencies. The new tests are consistent against a larger class of alternatives than the previous linear-time tests based on the (non-smoothed) empirical characteristic functions, while being much faster than the current state-of-the-art quadratic-time kernel-based or energy distance-based tests. Experiments on artificial benchmarks and on challenging real-world testing problems demonstrate that our tests give a better power/time tradeoff than competing approaches, and in some cases, better outright power than even the most expensive quadratic-time tests. This performance advantage is retained even in high dimensions, and in cases where the difference in distributions is not observable with low order statistics.