Robust Hypothesis Testing with Kernel Uncertainty Sets
Robust Hypothesis Testing with Kernel Uncertainty Sets
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DOI：
10.1109/isit50566.2022.9834349
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Zhongchang Sun;Shaofeng Zou
中科院分区：
文献类型：
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作者：
Zhongchang Sun;Shaofeng Zou
本文研究了在零假设和备择假设下，分布在某些不确定性集中的稳健假设检验问题。不确定性集是以数据驱动的方式构建的，即，它们围绕着经验分布。利用再生核Hilbert空间中分布的核均值嵌入之间的距离作为不确定集的距离度量。贝叶斯设置进行了研究，其目标是最大限度地减少最坏情况下的错误概率。首先针对字母表有限的情况获得了最优检验。对于无限字母表的情况，提出了一个易于处理的近似来量化最坏情况下的错误概率，并进一步应用核平滑方法来设计测试，推广到看不见的样本。提出了一种启发式鲁棒核检验方法，并证明了该方法具有指数一致性。数值结果证明了所提出的测试的性能。
In this paper, the robust hypothesis testing problem is investigated, where under the null and the alternative hypotheses, the distributions are assumed to be in some uncertainty sets. The uncertainty sets are constructed in a data-driven manner, i.e., they are centered around empirical distributions. The distance between kernel mean embeddings of distributions in the reproducing kernel Hilbert space is used as the distance metric of uncertainty sets. The Bayesian setting is studied, where the goal is to minimize the worst-case error probability. An optimal test is firstly obtained for the case with a finite alphabet. For the case with an infinite alphabet, a tractable approximation is proposed to quantify the worst-case error probability, and a kernel smoothing method is further applied to design test that generalizes to unseen samples. A heuristic robust kernel test is also proposed and proved to be exponentially consistent. Numerical results are provided to demonstrate the performance of the proposed tests.