用核嵌入构造似然比检验，显著提升高维弱信号下检测能力。
Likelihood Ratio Tests by Kernel Gaussian Embedding
- 结合核均值与核协方差嵌入，构建高斯化概率测度的似然比统计量。
- 在零假设下统计量为0，备择假设下发散至无穷，实现0/∞判别律。
- 适用于高维、弱信号场景，对现有MMD方法有统一与扩展作用。
我们提出一种基于核的非参数两样本检验方法，联合使用核均值和核协方差嵌入。该方法利用最新研究结果：这些组合嵌入可将不同概率测度映射为核再生希尔伯特空间（RKHS）上的互不相交高斯测度。借助这一“测度分离现象”，我们构造出基于高斯嵌入间相对熵的检验统计量，本质上即为似然比。该似然比专门设计用于检测两个高斯测度的相等性与奇异性，满足“0/∞”法则——在原假设下为0，在备择假设下趋于无穷。针对有限样本实现，我们引入正则化版本，并通过置换法校准。理论证明了该检验的一致性，并在温和条件下建立了统一的势能保证。进一步讨论表明，本框架统一并扩展了基于谱正则化MMD的先前方法。在合成数据与真实数据上的实验显示，相比当前最优方法，该方法在高维及弱信号情形下显著提升了检验功效。
原文摘要 · Abstract (English)
We propose a novel kernel-based nonparametric two-sample test, employing the combined use of kernel mean and kernel covariance embedding. Our test builds on recent results showing how such combined embeddings map distinct probability measures to mutually singular Gaussian measures on the kernel's RKHS. Leveraging this ``separation of measure phenomenon", we construct a test statistic based on the relative entropy between the Gaussian embeddings, in effect the likelihood ratio. The likelihood ratio is specifically tailored to detect equality versus singularity of two Gaussians, and satisfies a ``$0/\infty$" law, in that it vanishes under the null and diverges under the alternative. To implement the test in finite samples, we introduce a regularised version, calibrated by way of permutation. We prove consistency, establish uniform power guarantees under mild conditions, and discuss how our framework unifies and extends prior approaches based on spectrally regularized MMD. Empirical results on synthetic and real data demonstrate remarkable gains in power compared to state-of-the-art methods, particularly in high-dimensional and weak-signal regimes.
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