# Exact Statistical Inference for the Wasserstein Distance by Selective Inference

arXiv: https://arxiv.org/abs/2109.14206 · HTML: https://mlsanity.com/paper/2109.14206/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 21 of 21)

- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Optimal Inference After Model Selection](https://mlsanity.com/paper/1410.2597/) (2014)
- [Exact Post-Selection Inference for Sequential Regression Procedures](https://mlsanity.com/paper/1401.3889/) (2014)
- [Valid Inference Corrected for Outlier Removal](https://mlsanity.com/paper/1711.10635/) (2017)
- [Computing Valid p-value for Optimal Changepoint by Selective Inference using Dynamic Programming](https://mlsanity.com/paper/2002.09132/) (2020)
- [More Powerful Conditional Selective Inference for Generalized Lasso by Parametric Programming](https://mlsanity.com/paper/2105.04920/) (2021)
- [Computing Valid p-values for Image Segmentation by Selective Inference](https://mlsanity.com/paper/1906.00629/) (2019)
- [More powerful post-selection inference, with application to the Lasso](https://mlsanity.com/paper/1801.09037/) (2018)
- [Selective Inference for Group-Sparse Linear Models](https://mlsanity.com/paper/1607.08211/) (2016)
- [Post-Selection Inference for Changepoint Detection Algorithms with Application to Copy Number Variation Data](https://mlsanity.com/paper/1812.03644/) (2018)
- [Selecting the number of principal components: estimation of the true rank of a noisy matrix](https://mlsanity.com/paper/1410.8260/) (2014)
- [A significance test for forward stepwise model selection](https://mlsanity.com/paper/1405.3920/) (2014)
- [Selective Sequential Model Selection](https://mlsanity.com/paper/1512.02565/) (2015)
- [Conditional Selective Inference for Robust Regression and Outlier Detection using Piecewise-Linear Homotopy Continuation](https://mlsanity.com/paper/2104.10840/) (2021)
- [Fast and More Powerful Selective Inference for Sparse High-order Interaction Model](https://mlsanity.com/paper/2106.04929/) (2021)
- [Selective inference in regression models with groups of variables](https://mlsanity.com/paper/1511.01478/) (2015)
- [Integrative Methods for Post-Selection Inference Under Convex Constraints](https://mlsanity.com/paper/1605.08824/) (2016)
- [On parameter estimation with the Wasserstein distance](https://mlsanity.com/paper/1701.05146/) (2017)
- [More Powerful and General Selective Inference for Stepwise Feature Selection using the Homotopy Continuation Approach](https://mlsanity.com/paper/2012.13545/) (2020)

## Built on by (most cited first, 7 of 7)

- [Valid and Exact Statistical Inference for Multi-dimensional Multiple Change-Points by Selective Inference](https://mlsanity.com/paper/2110.08989/) (2021)
- [CAD-DA: Controllable Anomaly Detection after Domain Adaptation by Statistical Inference](https://mlsanity.com/paper/2310.14608/) (2023)
- [Statistical Inference for Autoencoder-based Anomaly Detection after Representation Learning-based Domain Adaptation](https://mlsanity.com/paper/2508.07049/) (2025)
- [Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm](https://mlsanity.com/paper/2303.07287/) (2023)
- [Bounded P-values in Parametric Programming-based Selective Inference](https://mlsanity.com/paper/2307.11351/) (2023)
- [Statistical Inference for the Dynamic Time Warping Distance, with Application to Abnormal Time-Series Detection](https://mlsanity.com/paper/2202.06593/) (2022)
- [Statistical Inference for Clustering-based Anomaly Detection](https://mlsanity.com/paper/2504.18633/) (2025)

## Abstract (arXiv)

In this paper, we study statistical inference for the Wasserstein distance, which has attracted much attention and has been applied to various machine learning tasks. Several studies have been proposed in the literature, but almost all of them are based on asymptotic approximation and do not have finite-sample validity. In this study, we propose an exact (non-asymptotic) inference method for the Wasserstein distance inspired by the concept of conditional Selective Inference (SI). To our knowledge, this is the first method that can provide a valid confidence interval (CI) for the Wasserstein distance with finite-sample coverage guarantee, which can be applied not only to one-dimensional problems but also to multi-dimensional problems. We evaluate the performance of the proposed method on both synthetic and real-world datasets.