Zero-Flow Two-Sample Tests

2026-07-23

AI Summary

This study addresses the two-sample testing problem of determining whether two sets of samples originate from the same underlying distribution. The authors propose a novel statistical discrepancy measure and associated two-sample test grounded in the null-flux principle: by learning local misalignment directional patterns to construct a witness function, the approach decouples witness function learning from hypothesis testing, thereby balancing model flexibility with accurate calibration of Type I error rates. The framework accommodates both regression-based and power-maximizing learning strategies. Empirical evaluations demonstrate that the method achieves high test power against structured distributional discrepancies on both synthetic and image data while rigorously controlling the Type I error rate.

Abstract

We propose a new approach to two-sample testing for deciding whether two sets of samples are drawn from the same distribution. The test is built on a statistical discrepancy based on the zero-flow criterion, termed zero-flow discrepancy (ZFD). We prove the validity of ZFD and propose a practical testing procedure, termed the zero-flow two-sample test (ZF2ST). The key idea is to learn how samples from the two distributions are locally misaligned and use the resulting directional pattern as evidence of distributional difference. By separating witness learning from hypothesis evaluation, ZF2ST can use flexible neural networks while maintaining valid statistical calibration. We develop both regression-based and power-maximized approaches for learning the witness. Experiments on synthetic and image datasets demonstrate that ZF2ST can achieve strong testing power for structured distributional changes while maintaining well-calibrated type-I error.

Problem

• two-sample testing
• distributional difference
• statistical discrepancy
• type-I error calibration
• testing power

Innovation

• zero-flow discrepancy
• two-sample test
• witness function
• neural network calibration
• distributional difference