Spectral regularized kernel two-sample tests
谱正则化核两样本检验AI 解读 ↓
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谱正则化核两样本检验
- 研究问题
- 研究基于概率分布 RKHS 嵌入的两样本检验是否具有最优分离边界，并探讨如何改进 MMD 检验及实现正则参数的自适应选择。
- 方法与思路
- 作者将协方差信息通过谱正则化引入 MMD 检验，构造数据驱动选择正则参数的自适应版本，并研究以样本置换确定阈值的版本。
- 材料陈述的贡献
- 摘要称，新检验在所研究框架下达到极小极大最优，具有更小的 Hellinger 分离边界；自适应版本相差对数因子，并报告数值比较优势。
- 阅读关注点
- 应核对备择分布类、核与谱条件、Hellinger 分离边界的定义，以及自适应和置换结论的条件，并检查实验比较的公平性与范围。
- 阅读边界
- 仅据版本未注明的索引摘要；最优性及性能优势依赖未展开的理论框架和实验设置，不能推断其对所有分布与核均成立。
原文摘要
Over the last decade, an approach that has gained a lot of popularity to tackle nonparametric testing problems on general (i.e., non-Euclidean) domains is based on the notion of reproducing kernel Hilbert space (RKHS) embedding of probability distributions. The main goal of our work is to understand the optimality of two-sample tests constructed based on this approach. First, we show the popular MMD (maximum mean discrepancy) two-sample test to be not optimal in terms of the separation boundary measured in Hellinger distance. Second, we propose a modification to the MMD test based on spectral regularization by taking into account the covariance information (which is not captured by the MMD test) and prove the proposed test to be minimax optimal with a smaller separation boundary than that achieved by the MMD test. Third, we propose an adaptive version of the above test which involves a data-driven strategy to choose the regularization parameter and show the adaptive test to be almost minimax optimal up to a logarithmic factor. Moreover, our results hold for the permutation variant of the test where the test threshold is chosen elegantly through the permutation of the samples. Through numerical experiments on synthetic and real data, we demonstrate the superior performance of the proposed test in comparison to the MMD test and other popular tests in the literature.