Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case
Abstract
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.
Summary
This paper addresses the problem of extracting invariant features from data to improve prediction accuracy in scenarios where confounding variables affect both the predictors (X) and the response variable (Y). The core idea is to find a feature representation W = f(X) that predicts Y well but is independent of the confounding variables (Z) conditioned on Y. This conditional independence ensures that W captures the true relationship between X and Y, rather than spurious correlations caused by Z. The authors propose a novel methodology based on the Optimal Transport Barycenter Problem (OTBP) to enforce conditional invariance. They replace the conditional independence requirement (W ⊥⊥ Z | Y) with a more tractable independence condition (W ⊥⊥ Z_Y), where Z_Y is the solution to the OTBP for Z given Y. They demonstrate that this relaxation is equivalent to the original constraint in the Gaussian case. When the true confounders Z are unknown, they suggest using measurable contextual variables S as surrogates, which maintains equivalence in the Gaussian case under certain conditions. The paper focuses on the Gaussian case, deriving a closed-form solution for a linear feature extractor based on the eigenvectors of a known matrix. The authors validate their method through simulated experiments and compare it with Anchor Regression, demonstrating improved performance in situations with distributional shifts. The significance of this work lies in its ability to extract features robust to environmental changes without requiring knowledge of the underlying causal model or all confounding variables. The use of OTBP for enforcing conditional independence is a novel contribution, providing a general framework for invariant feature extraction that can be extended to non-Gaussian distributions and non-linear feature extractors, as the authors note for future work.
Key Insights
- •The paper introduces a novel method for invariant feature extraction based on penalizing the dependence between extracted features and confounding variables conditioned on the response variable.
- •The core of the methodology relies on replacing the conditional independence requirement (W ⊥⊥ Z | Y) with a more readily implementable plain independence condition (W ⊥⊥ Z_Y), where Z_Y is the solution to the Optimal Transport Barycenter Problem (OTBP) for Z given Y. This replacement is shown to be equivalent in the Gaussian case.
- •When true confounders (Z) are unknown, the authors demonstrate that measurable contextual variables (S) can be used as surrogates without relaxation in the Gaussian case, provided the covariance matrix Σ_{ZS} has full rank.
- •For the Gaussian case, the feature extractor W is a linear function of X, and the solution is found in closed form as the eigenvectors of a matrix H = (1-λ)CC' - λDD', where C = Σ_{XY}, D = Σ_{XS_Y}, and λ is a regularization parameter.
- •The paper provides a theoretical justification for the proposed methodology, including lemmas and theorems demonstrating the equivalence between conditional and plain independence under Gaussian assumptions and the optimality of the closed-form solution.
- •Experimental results show that the proposed barycentric method outperforms Anchor Regression and ordinary least squares (OLS) in scenarios with distributional shifts.
- •The authors acknowledge that the framework is currently limited to the Gaussian case and linear feature extractors, but they suggest that the methodology can be extended to more general distributions and non-linear feature extractors in future work.
Practical Implications
- •The proposed methodology can be applied in various domains where predictions need to be robust to environmental changes, such as medical diagnosis, where algorithms often fail to generalize well to populations different from the training data.
- •Researchers and engineers can use the closed-form solution derived in the paper to implement a linear feature extractor that minimizes the influence of confounding variables on predictions, particularly in Gaussian settings.
- •The methodology provides a framework for incorporating domain knowledge in the form of contextual variables (S) to improve the robustness of predictive models.
- •Future research directions include extending the methodology to non-Gaussian distributions, non-linear feature extractors, and scenarios with multiple source environments with potentially different observed features.
- •The paper opens up opportunities for exploring the use of Optimal Transport Barycenters in other machine learning tasks involving invariance and domain adaptation.