Nonparametric Detection of Anomalous Data via Kernel Mean Embedding

An anomaly detection problem is investigated, in which there are totally n sequences with s anomalous sequences to be detected. Each normal sequence contains m independent and identically distributed (i.i.d.) samples drawn from a distribution p, whereas each anomalous sequence contains m i.i.d. samples drawn from a distribution q that is distinct from p. The distributions p and q are assumed to be unknown a priori. Two scenarios, respectively with and without a reference sequence generated by p, are studied. Distribution-free tests are constructed using maximum mean discrepancy (MMD) as the metric, which is based on mean embeddings of distributions into a reproducing kernel Hilbert space (RKHS). For both scenarios, it is shown that as the number n of sequences goes to infinity, if the value of s is known, then the number m of samples in each sequence should be at the order O(log n) or larger in order for the developed tests to consistently detect s anomalous sequences. If the value of s is unknown, then m should be at the order strictly larger than O(log n). Computational complexity of all developed tests is shown to be polynomial. Numerical results demonstrate that our tests outperform (or perform as well as) the tests based on other competitive traditional statistical approaches and kernel-based approaches under various cases. Consistency of the proposed test is also demonstrated on a real data set.