Testing Conditional Independence via the Spectral Generalized Covariance Measure: Beyond Euclidean Data

We propose a conditional independence (CI) test based on a new measure, the \emph{spectral generalized covariance measure} (SGCM). The SGCM is constructed by approximating the basis expansion of the squared norm of the conditional cross-covariance operator, using data-dependent bases obtained via spectral decompositions of empirical covariance operators. This construction avoids direct estimation of conditional mean embeddings and reduces the problem to scalar-valued regressions, resulting in robust finite-sample size control. Theoretically, we derive the limiting distribution of the SGCM statistic, establish the validity of a wild bootstrap for inference, and obtain uniform asymptotic size control under doubly robust conditions. As an additional contribution, we show that exponential kernels induced by continuous semimetrics of negative type are characteristic on general Polish spaces -- with extensions to finite tensor products -- thereby providing a foundation for applying our test and other kernel methods to complex objects such as distribution-valued data and curves on metric spaces. Extensive simulations indicate that the SGCM-based CI test attains near-nominal size and exhibits power competitive with or superior to state-of-the-art alternatives across a range of challenging scenarios.