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 spectral generalized covariance measure (SGCM). The SGCM is constructed by expressing the squared norm of the conditional cross-covariance operator in spectral coordinates and approximating it in finite dimensions using data-dependent bases obtained from empirical covariance operators. This avoids direct estimation of conditional mean embeddings and reduces nuisance estimation to a finite collection of scalar-valued regressions. On the theoretical side, under a doubly robust product-bias condition, we establish uniform bootstrap validity and uniform asymptotic size control, and derive nontrivial uniform power and uniform consistency over classes of projected separated alternatives. The analysis also clarifies the role of spectral truncation: stronger truncation relaxes nuisance-estimation requirements, whereas weaker truncation retains more of the projected signal. To support applications beyond Euclidean data, we develop characteristic-kernel constructions on general Polish spaces via a pullback principle and non-constant completely monotone transforms of continuous negative-type semimetrics, with closure under finite tensor products. These constructions cover examples such as distribution-valued data, curves in metric spaces, and manifold-valued observations. Simulations show near-nominal size in the main settings and competitive power across a range of challenging scenarios.