# Integral Probability Metrics Meet Neural Networks: The Radon-Kolmogorov-Smirnov Test

arXiv: https://arxiv.org/abs/2309.02422 · HTML: https://mlsanity.com/paper/2309.02422/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 5 Sep 2023.

## Builds on (most cited first, 14 of 14)

- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Learning Transferable Features with Deep Adaptation Networks](https://mlsanity.com/paper/1502.02791/) (2015)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Sobolev GAN](https://mlsanity.com/paper/1711.04894/) (2017)
- [The Falling Factorial Basis and Its Statistical Applications](https://mlsanity.com/paper/1405.0558/) (2014)
- [Principal Differences Analysis: Interpretable Characterization of Differences between Distributions](https://mlsanity.com/paper/1510.08956/) (2015)
- [A Higher-Order Kolmogorov-Smirnov Test](https://mlsanity.com/paper/1903.10083/) (2019)
- [Direction-Projection-Permutation for High Dimensional Hypothesis Tests](https://mlsanity.com/paper/1304.0796/) (2013)

## Built on by (most cited first, 2 of 2)

- [Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance](https://mlsanity.com/paper/2504.11299/) (2025)
- [A nonparametric two-sample test using a parametric integral probability metric](https://mlsanity.com/paper/2606.16941/) (2026)

## Abstract (arXiv)

Integral probability metrics (IPMs) constitute a general class of nonparametric two-sample tests that are based on maximizing the mean difference between samples from one distribution P versus another Q, over all choices of data transformations f living in some function space F. Inspired by recent work that connects what are known as functions of textitRadon bounded variation (RBV) and neural networks (Parhi and Nowak, 2021, 2023), we study the IPM defined by taking F to be the unit ball in the RBV space of a given smoothness degree k ≥ 0. This test, which we refer to as the textitRadon-Kolmogorov-Smirnov (RKS) test, can be viewed as a generalization of the well-known and classical Kolmogorov-Smirnov (KS) test to multiple dimensions and higher orders of smoothness. It is also intimately connected to neural networks: we prove that the witness in the RKS test -- the function f achieving the maximum mean difference -- is always a ridge spline of degree k, i.e., a single neuron in a neural network. We can thus leverage the power of modern neural network optimization toolkits to (approximately) maximize the criterion that underlies the RKS test. We prove that the RKS test has asymptotically full power at distinguishing any distinct pair P not= Q of distributions, derive its asymptotic null distribution, and carry out experiments to elucidate the strengths and weaknesses of the RKS test versus the more traditional kernel MMD test.