We propose a conditional independence (CI) test based on a new measure, the \emph{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.

翻译：我们提出一种基于新测度——谱广义协方差测度（SGCM）的条件独立性检验方法。SGCM通过将条件互协方差算子的平方范数在谱坐标下展开，并利用经验协方差算子获得的数据依赖基在有限维空间中进行近似构建。该方法避免了直接估计条件均值嵌入，将干扰项估计简化为有限个标量回归问题。在理论层面，通过双重稳健乘积偏差条件，我们建立了自助法的均匀有效性、渐近均匀尺寸控制，并推导出投影分离备择假设类别下的非平凡均匀检验力和一致性。研究同时阐明了谱截断的作用：强截断可放宽干扰项估计要求，而弱截断则保留更多投影信号。为拓展至欧几里得数据以外的应用场景，我们基于拉回原理和连续负定半度量上的非恒定完全单调变换，发展了广义波兰空间中的特征核函数构造方法，且该构造在有限张量积下保持封闭性。这些框架可处理分布型数据、度量空间中的曲线以及流形观测值等案例。仿真结果表明，该检验在主设场景中能达到近标称尺寸，并在多种挑战性情境下展现出具有竞争力的检验力。