# Quantile-constrained Wasserstein projections for robust interpretability of numerical and machine learning models

arXiv: https://arxiv.org/abs/2209.11539 · HTML: https://mlsanity.com/paper/2209.11539/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 132 references that are in the map.
- First posted on arXiv 23 Sep 2022.

## Builds on (most cited first, 30 of 34)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Universal adversarial perturbations](https://mlsanity.com/paper/1610.08401/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Generalizing to Unseen Domains: A Survey on Domain Generalization](https://mlsanity.com/paper/2103.03097/) (2021)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Explaining Explanations: An Overview of Interpretability of Machine Learning](https://mlsanity.com/paper/1806.00069/) (2018)
- [Towards Out-Of-Distribution Generalization: A Survey](https://mlsanity.com/paper/2108.13624/) (2021)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Distribution Calibration for Regression](https://mlsanity.com/paper/1905.06023/) (2019)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [Look at the Variance! Efficient Black-box Explanations with Sobol-based Sensitivity Analysis](https://mlsanity.com/paper/2111.04138/) (2021)
- [Correlation and variable importance in random forests](https://mlsanity.com/paper/1310.5726/) (2013)
- [Statistical stability indices for LIME: obtaining reliable explanations for Machine Learning models](https://mlsanity.com/paper/2001.11757/) (2020)
- [Generalization bounds for deep learning](https://mlsanity.com/paper/2012.04115/) (2020)
- [Adversarial Risk via Optimal Transport and Optimal Couplings](https://mlsanity.com/paper/1912.02794/) (2019)
- [Transport-based Counterfactual Models](https://mlsanity.com/paper/2108.13025/) (2021)
- [Prior knowledge elicitation: The past, present, and future](https://mlsanity.com/paper/2112.01380/) (2021)
- [SHAFF: Fast and consistent SHApley eFfect estimates via random Forests](https://mlsanity.com/paper/2105.11724/) (2021)
- [Variable selection for Gaussian processes via sensitivity analysis of the posterior predictive distribution](https://mlsanity.com/paper/1712.08048/) (2017)
- [Deep Learning Generalization and the Convex Hull of Training Sets](https://mlsanity.com/paper/2101.09849/) (2021)
- [MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA](https://mlsanity.com/paper/2102.13347/) (2021)
- [Sampling, Intervention, Prediction, Aggregation: A Generalized Framework for Model-Agnostic Interpretations](https://mlsanity.com/paper/1904.03959/) (2019)
- [Explaining Machine Learning Models using Entropic Variable Projection](https://mlsanity.com/paper/1810.07924/) (2018)
- [Sensitivity analysis for Bayesian hierarchical models](https://mlsanity.com/paper/1312.4797/) (2013)

## Built on by (most cited first, 4 of 4)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Unveil Sources of Uncertainty: Feature Contribution to Conformal Prediction Intervals](https://mlsanity.com/paper/2505.13118/) (2025)
- [Wasserstein-Cram'er-Rao Theory of Unbiased Estimation](https://mlsanity.com/paper/2511.07414/) (2025)
- [Evaluating Black-Box Vulnerabilities with Wasserstein-Constrained Data Perturbations](https://mlsanity.com/paper/2603.15867/) (2026)

## Abstract (arXiv)

Robustness studies of black-box models is recognized as a necessary task for numerical models based on structural equations and predictive models learned from data. These studies must assess the model's robustness to possible misspecification of regarding its inputs (e.g., covariate shift). The study of black-box models, through the prism of uncertainty quantification (UQ), is often based on sensitivity analysis involving a probabilistic structure imposed on the inputs, while ML models are solely constructed from observed data. Our work aim at unifying the UQ and ML interpretability approaches, by providing relevant and easy-to-use tools for both paradigms. To provide a generic and understandable framework for robustness studies, we define perturbations of input information relying on quantile constraints and projections with respect to the Wasserstein distance between probability measures, while preserving their dependence structure. We show that this perturbation problem can be analytically solved. Ensuring regularity constraints by means of isotonic polynomial approximations leads to smoother perturbations, which can be more suitable in practice. Numerical experiments on real case studies, from the UQ and ML fields, highlight the computational feasibility of such studies and provide local and global insights on the robustness of black-box models to input perturbations.