# Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case

arXiv: https://arxiv.org/abs/2512.20914 · HTML: https://mlsanity.com/paper/2512.20914/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 28 references that are in the map.
- First posted on arXiv 24 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [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)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Out of Distribution Generalization in Machine Learning](https://mlsanity.com/paper/2103.02667/) (2021)
- [Out-of-distribution Generalization in the Presence of Nuisance-Induced Spurious Correlations](https://mlsanity.com/paper/2107.00520/) (2021)
- [Domain Generalization via Optimal Transport with Metric Similarity Learning](https://mlsanity.com/paper/2007.10573/) (2020)
- [Environment Invariant Linear Least Squares](https://mlsanity.com/paper/2303.03092/) (2023)
- [Data Driven Conditional Optimal Transport](https://mlsanity.com/paper/1910.11422/) (2019)
- [Transfer Learning with Uncertainty Quantification: Random Effect Calibration of Source to Target (RECaST)](https://mlsanity.com/paper/2211.16557/) (2022)
- [Conditional Density Estimation, Latent Variable Discovery and Optimal Transport](https://mlsanity.com/paper/1910.14090/) (2019)
- [Transfer Learning under Group-Label Shift: A Semiparametric Exponential Tilting Approach](https://mlsanity.com/paper/2509.22268/) (2025)

## Abstract (arXiv)

A methodology is developed to extract d invariant features W=f(X) that predict a response variable Y without being confounded by variables Z that may influence both X and Y. The methodology's main ingredient is the penalization of any statistical dependence between W and Z conditioned on Y, replaced by the more readily implementable plain independence between W and the random variable ZY = T(Z,Y) that solves the [Monge] Optimal Transport Barycenter Problem for Zmid Y. In the Gaussian case considered in this article, the two statements are equivalent. When the true confounders Z are unknown, other measurable contextual variables S can be used as surrogates, a replacement that involves no relaxation in the Gaussian case if the covariance matrix SigmaZS has full range. The resulting linear feature extractor adopts a closed form in terms of the first d eigenvectors of a known matrix. The procedure extends with little change to more general, non-Gaussian / non-linear cases.