This could be an interesting paper by Stefanie Jegelka!
From the 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 under arbitrarily given group actions.
We tackle this problem by studying discrepancy-based measures of invariance that can capture even subtle distributional asymmetries.
Our first contribution is to show that, while detecting worst-case asymmetries can be computationally intractable, a randomized method can estimate closeness measures to invariance within universal constant factors.
This provides a general framework for statistical testing of invariance under compact group actions.
Despite the extensive and well-established literature on group-based testing, our methodology, to the best of our knowledge, is the first to provide statistical tests for general group invariances with finite-sample guarantees on Type II errors against worst-case alternatives.
We instantiate the framework for common probability discrepancies, including total variation, Wasserstein distances, integral probability metrics, energy distance, and maximum mean discrepancy, obtaining explicit sample-complexity guarantees from empirical convergence rates."
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