This paper provides a unifying view of optimal kernel hypothesis testing across the MMD two-sample, HSIC independence, and KSD goodness-of-fit frameworks. Minimax optimal separation rates in the kernel and $L^2$ metrics are presented, with two adaptive kernel selection methods (kernel pooling and aggregation), and under various testing constraints: computational efficiency, differential privacy, and robustness to data corruption. Intuition behind the derivation of the power results is provided in a unified way across the three frameworks, and open problems are highlighted.

翻译：本文为MMD双样本检验、HSIC独立性检验及KSD拟合优度检验框架下的最优核假设检验提供了统一的理论视角。研究给出了核度量与$L^2$度量下的极小极大最优分离速率，提出了两种自适应核选择方法（核池化与核聚合），并分析了多种检验约束条件：计算效率、差分隐私及数据污染鲁棒性。通过统一的理论框架阐释了检验功效结果背后的数学直觉，同时指出了若干有待解决的开放性问题。