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Two-sample testing decides whether two datasets are generated from the samedistribution. This paper studies variable selection for two-sample testing, thetask being to identify the variables (or dimensions) responsible for thediscrepancies between the two distributions. This task is relevant to manyproblems of pattern analysis and machine learning, such as dataset shiftadaptation, causal inference and model validation. Our approach is based on atwo-sample test based on the Maximum Mean Discrepancy (MMD). We optimise theAutomatic Relevance Detection (ARD) weights defined for individual variables tomaximise the power of the MMD-based test. For this optimisation, we introducesparse regularisation and propose two methods for dealing with the issue ofselecting an appropriate regularisation parameter. One method determines theregularisation parameter in a data-driven way, and the other aggregates theresults of different regularisation parameters. We confirm the validity of theproposed methods by systematic comparisons with baseline methods, anddemonstrate their usefulness in exploratory analysis of high-dimensionaltraffic simulation data. Preliminary theoretical analyses are also provided,including a rigorous definition of variable selection for two-sample testing.

Variable Selection in Maximum Mean Discrepancy for Interpretable Distribution Comparison