Spectral Representation Learning for Conditional Independence Testing

Conditional independence (CI) is fundamental to causal inference, feature selection, and graphical modeling, yet reliable testing remains challenging in high-dimensional settings. We investigate whether representation learning can address this limitation. Specifically, we focus on representations derived from the singular value decomposition of partial covariance operators and use them to construct a simple test statistic, reminiscent of the Hilbert-Schmidt Independence Criterion (HSIC). We also introduce a bi-level contrastive algorithm to learn these representations. Our theory links representation learning error to test performance and establishes asymptotic validity and power guarantees. Experiments on real and synthetic data suggest that this approach offers a principled and statistically grounded path toward scalable CI testing, bridging classical kernel-based CI tests with modern representation learning.