Variable Selection in Maximum Mean Discrepancy for Interpretable
  Distribution Comparison
Kensuke Mitsuzawa Motonobu Kanagawa Stefano Bortoli Margherita Grossi Paolo Papotti
Two-sample testing decides whether two datasets are generated from the same
distribution. This paper studies variable selection for two-sample testing, the
task being to identify the variables (or dimensions) responsible for the
discrepancies between the two distributions. This task is relevant to many
problems of pattern analysis and machine learning, such as dataset shift
adaptation, causal inference and model validation. Our approach is based on a
two-sample test based on the Maximum Mean Discrepancy (MMD). We optimise the
Automatic Relevance Detection (ARD) weights defined for individual variables to
maximise the power of the MMD-based test. For this optimisation, we introduce
sparse regularisation and propose two methods for dealing with the issue of
selecting an appropriate regularisation parameter. One method determines the
regularisation parameter in a data-driven way, and the other aggregates the
results of different regularisation parameters. We confirm the validity of the
proposed methods by systematic comparisons with baseline methods, and
demonstrate their usefulness in exploratory analysis of high-dimensional
traffic simulation data. Preliminary theoretical analyses are also provided,
including a rigorous definition of variable selection for two-sample testing.