Integral Probability Metrics

Also known as IPM

An integral probability metric (IPM) measures how distinguishable two distributions are to a chosen family of functions, so that family determines the distance’s meaning and sensitivity. Common choices include bounded indicators, Lipschitz functions for Wasserstein distance, kernels for maximum mean discrepancy, and learned or structured distinguishers. IPMs support two-sample and goodness-of-fit testing, generative-model evaluation, out-of-distribution detection, uncertainty quantification, and Bayesian experimental design; theory also studies their estimation, computational cost, and robustness to model error. A central tension is that a test class must balance statistical stability and tractability against the power to detect relevant differences, including rare-event shifts.

Latest papers

15 papers2 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Integral Probability Boundaries
Oct 27, 2026Bogdan Kulynych, Shahab AsoodehIntegral Probability Metrics Generalization Bounds
Biomedical Data Science Center, Lausanne University Hospital (CHUV) · McMaster University and Vector Institute
How Geometry Shapes Generative Model Evaluation on Riemannian Manifolds
Oct 24, 2026Misha Oberski, Valentijn Oldenburg, Xanthos Kekkos +6Generative AI Evaluation Wasserstein Distance
University of Amsterdam · Independent Researcher · AMLab, University of Amsterdam
Differentially Private Synthetic Data via Generalized Moment Matching
Oct 21, 2026Rundong Ding, Yiyun He, Yizhe ZhuDifferentially Private Synthetic Data Generation Differential Privacy
Department of Mathematics, University of Southern California · Department of Mathematics, University of California San Diego
Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
Sep 19, 2026Mehrdad MohammadiIntegral Probability Metrics Wasserstein Distance
Department of Statistics University of Illinois Urbana-Champaign Champaign, IL 61820, USA
A Probabilistic Circuit-Induced Pseudo-Metric for Out-of-Distribution Detection
Aug 10, 2026Bhumika K, Vidhya S, Narayanan C KrishnanProbabilistic Circuits OOD Detection
Mehta Family School of Data Science and Artificial Intelligence Indian Institute of Technology Palakkad Palakkad, Kerala, India
Testing Distributions Against Bounded Distinguishers
Jul 17, 2026Mark Bun, Rathin Desai, Renato Ferreira PintoIntegral Probability Metrics
Boston University · Columbia University
A nonparametric two-sample test using a parametric integral probability metric
Jun 15, 2026Yuha Park, Yongdai KimTwo-Sample Testing Statistical Learning Theory
Department of Mathematics, University of Hamburg, Hamburg, Germany. · Department of Statistics, Seoul National University, Seoul, Republic of Korea.
Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space
May 17, 2026Kelvin Kan, Xingjian Li, Benjamin J. Zhang +3Integral Probability Metrics Diffusion Models
Department of Mathematics Oden Institute UCLA · Department of Mathematics University of Texas at Austin · Rutgers University +3
Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale
May 13, 2026Shashaank Aiyer, Yishay Mansour, Shay Moran +2Agnostic Learning Statistical Learning Theory
University of Maryland · Tel Aviv University and Google Research · Technion and Google Research +1
Beyond Expected Information Gain: Stable Bayesian Optimal Experimental Design with Integral Probability Metrics and Plug-and-Play Extensions
Apr 23, 2026Di Wu, Ling Liang, Haizhao YangBayesian Experimental Design Integral Probability Metrics
Department of Mathematics The University of Tennessee, Knoxville TN 37916, USA · Department of Mathematics and Department of Computer Science University of Maryland, College Park MD 20742, USA
A Theoretical Framework for Statistical Evaluability of Generative Models
Apr 7, 2026Shashaank Aiyer, Yishay Mansour, Shay Moran +1Generative AI Evaluation Statistical Learning Theory
University of Maryland · Tel Aviv University and Google Research · Technion and Google Research
Quantification of Credal Uncertainty: A Distance-Based Approach
Mar 28, 2026Xabier Gonzalez-Garcia, Siu Lun Chau, Julian Rodemann +6Uncertainty Quantification Aleatoric Uncertainty
Dept. of Statistics, CS & Mathematics, Public Univ. of Navarre · EPIC Lab, Nanyang Technological Univ. Singapore · Rational Intelligence Lab, CISPA Helmholtz Center for Information Security +7
Scalable Kernel-Based Distances for Statistical Inference and Integration
Feb 25, 2026Masha NaslidnykNumerical Integration Maximum Mean Discrepancy
Department of Computer Science University College London
Orlicz-Sobolev with Musielak: An Efficient Regularization Approach for Graph-based IPM
Oct 29, 2025Tam Le, Truyen Nguyen, Hideitsu Hino +1Integral Probability Metrics Optimal Transport
University of Akron
Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance
Apr 15, 2025Peter Matthew Jacobs, Foad Namjoo, Jeff M. PhillipsTwo-Sample Testing Distribution Matching
University of Wisconsin–Madison, USA · University of Utah, USA

Oct 27, 2026J/K move · Enter open · S save

Integral Probability Boundaries

Bogdan Kulynych, Shahab Asoodeh

Biomedical Data Science Center, Lausanne University Hospital (CHUV) · McMaster University and Vector Institute

Integral Probability Metrics Generalization Bounds

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Bounds based on Integral Probability Metrics (IPMs) are ubiquitous in machine learning and statistics, appearing, e.g., in generalization, alignment, fairness, and privacy. IPMs reduce the discrepancy between two distributions to a single worst-case gap between expectations. This compression is costly: for rare events, safety failures, or under-represented groups, IPM-based bounds are loose or vacuous. We introduce a non-linear generalization of IPMs, which we term Integral Probability Boundaries (IPBs). Unlike IPMs, IPBs provide bounds that are tight at the extremes. We derive their basic properties and present algorithms for estimating them in practically relevant settings. We demonstrate the usage of IPBs in both theoretical and practical applications for alignment, judge-free performance estimation, and fairness, where they enable us to obtain sharp bounds on performance and fairness gaps, e.g., 15 p.p. more precise estimates on fairness gaps for rare groups in a realistic generative modeling task, compared to the state of the art.