# ReLU integral probability metric and its applications

arXiv: https://arxiv.org/abs/2504.18897 · HTML: https://mlsanity.com/paper/2504.18897/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 55 of its 112 references that are in the map.
- First posted on arXiv 26 Apr 2025.

## Builds on (most cited first, 30 of 55)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [A Reductions Approach to Fair Classification](https://mlsanity.com/paper/1803.02453/) (2018)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [AI Fairness 360: An Extensible Toolkit for Detecting, Understanding, and Mitigating Unwanted Algorithmic Bias](https://mlsanity.com/paper/1810.01943/) (2018)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Empirical Risk Minimization under Fairness Constraints](https://mlsanity.com/paper/1802.08626/) (2018)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [A Survey on Causal Inference](https://mlsanity.com/paper/2002.02770/) (2020)
- [Censoring Representations with an Adversary](https://mlsanity.com/paper/1511.05897/) (2015)
- [Causal Inference in Natural Language Processing: Estimation, Prediction, Interpretation and Beyond](https://mlsanity.com/paper/2109.00725/) (2021)
- [Fair Regression: Quantitative Definitions and Reduction-based Algorithms](https://mlsanity.com/paper/1905.12843/) (2019)
- [A survey on datasets for fairness-aware machine learning](https://mlsanity.com/paper/2110.00530/) (2021)
- [Rethinking Rotated Object Detection with Gaussian Wasserstein Distance Loss](https://mlsanity.com/paper/2101.11952/) (2021)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [Mind the Class Weight Bias: Weighted Maximum Mean Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1705.00609/) (2017)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)
- [Controllable Invariance through Adversarial Feature Learning](https://mlsanity.com/paper/1705.11122/) (2017)
- [Fair Mixup: Fairness via Interpolation](https://mlsanity.com/paper/2103.06503/) (2021)
- [Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competition](https://mlsanity.com/paper/1707.02641/) (2017)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [Fairness in Reinforcement Learning](https://mlsanity.com/paper/1611.03071/) (2016)
- [Wasserstein Dependency Measure for Representation Learning](https://mlsanity.com/paper/1903.11780/) (2019)
- [Balancing Covariates via Propensity Score Weighting](https://mlsanity.com/paper/1609.07494/) (2016)

## Built on by (most cited first, 1 of 1)

- [A nonparametric two-sample test using a parametric integral probability metric](https://mlsanity.com/paper/2606.16941/) (2026)

## Abstract (arXiv)

We propose a parametric integral probability metric (IPM) to measure the discrepancy between two probability measures. The proposed IPM leverages a specific parametric family of discriminators, such as single-node neural networks with ReLU activation, to effectively distinguish between distributions, making it applicable in high-dimensional settings. By optimizing over the parameters of the chosen discriminator class, the proposed IPM demonstrates that its estimators have good convergence rates and can serve as a surrogate for other IPMs that use smooth nonparametric discriminator classes. We present an efficient algorithm for practical computation, offering a simple implementation and requiring fewer hyperparameters. Furthermore, we explore its applications in various tasks, such as covariate balancing for causal inference and fair representation learning. Across such diverse applications, we demonstrate that the proposed IPM provides strong theoretical guarantees, and empirical experiments show that it achieves comparable or even superior performance to other methods.