Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910178575646720 |
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| author | Bounos, Ian Groisman, Pablo Sued, Mariela Tabak, Esteban |
| author_facet | Bounos, Ian Groisman, Pablo Sued, Mariela Tabak, Esteban |
| contents | 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 $Z_Y = T(Z,Y)$ that solves the [Monge] Optimal Transport Barycenter Problem for $Z\mid Y$. 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 $Σ_{ZS}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case Bounos, Ian Groisman, Pablo Sued, Mariela Tabak, Esteban Statistics Theory Applications Machine Learning 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 $Z_Y = T(Z,Y)$ that solves the [Monge] Optimal Transport Barycenter Problem for $Z\mid Y$. 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 $Σ_{ZS}$ 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. |
| title | Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case |
| topic | Statistics Theory Applications Machine Learning |
| url | https://arxiv.org/abs/2512.20914 |