A Robust Kernel Statistical Test of Invariance: Detecting Subtle Asymmetries

Ashkan Soleymani, Behrooz Tahmasebi, Stefanie Jegelka, Patrick Jaillet

Keywords: Invariance, Hypothesis Testing, Kernel Methods

Abstract: While invariances naturally arise in almost any type of real-world data, no efficient and robust test exists for detecting them in observational data for arbitrarily given group actions. We tackle this problem by studying measures of invariance that can capture even negligible underlying patterns. Our first contribution is to show that, while detecting subtle asymmetries is computationally intractable, a randomized method can be used to estimate robust closeness measures to invariance within constant factors. This provides a general framework for robust statistical tests of invariance. In addition, we focus on kernel methods and propose deterministic algorithms for robust testing with respect to both finite and infinite groups, accom- panied by a rigorous analysis of their convergence rates and sample complexity. Finally, we revisit the general framework in the specific case of kernel methods, showing that recent closeness measures to invariance, defined via group averaging, are provably robust, leading to powerful randomized algorithms.

Submission Number: 8