We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the \emph{equality} of two non-atomic (Borel) probability measures on a locally compact Polish space is \emph{equivalent} to testing for the \emph{singularity} between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an information-theoretic perspective than non-parametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This ``separation of measure phenomenon'' appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods.

翻译：我们证明了核协方差嵌入能够实现不同连续概率分布在信息论意义上的完美分离。从统计学角度而言，我们证明了在局部紧波兰空间上检验两个非原子（Borel）概率测度的相等性，等价于在再生核希尔伯特空间中检验两个中心化高斯测度之间的奇异性。对应的高斯测度通过概率测度的核协方差嵌入概念定义，而希尔伯特空间则由嵌入核生成。从信息论视角来看，区分奇异高斯测度在结构上比非参数双样本检验更为简单，尤其在复杂或高维领域中。这是因为奇异高斯测度本质上支撑在分离的仿射子空间上。我们的证明利用了经典的Feldman-Hájek二分定理，并表明即使对连续分布进行微小扰动，也会通过其高斯嵌入被极大程度地放大。这种“测度分离现象”似乎是通过嵌入实现的无限维优势，有望为广泛领域内高效推断工具的设计提供理论依据。该现象的揭示也以精确而简洁的数学表述，凝练了支撑核方法实证有效性的核心机制。