We introduce Isolation Distributional Kernel as a new way to measure the similarity between two distributions. Existing approaches based on kernel mean embedding, which convert a point kernel to a distributional kernel, have two key issues: the point kernel employed has a feature map with intractable dimensionality; and it is {\em data independent}. This paper shows that Isolation Distributional Kernel (IDK), which is based on a {\em data dependent} point kernel, addresses both key issues. We demonstrate IDK's efficacy and efficiency as a new tool for kernel based anomaly detection for both point and group anomalies. Without explicit learning, using IDK alone outperforms existing kernel based point anomaly detector OCSVM and other kernel mean embedding methods that rely on Gaussian kernel. For group anomaly detection,we introduce an IDK based detector called IDK$^2$. It reformulates the problem of group anomaly detection in input space into the problem of point anomaly detection in Hilbert space, without the need for learning. IDK$^2$ runs orders of magnitude faster than group anomaly detector OCSMM.We reveal for the first time that an effective kernel based anomaly detector based on kernel mean embedding must employ a characteristic kernel which is data dependent.

A recent Wasserstein Weisfeiler-Lehman (WWL) Graph Kernel has a distinctive feature: Representing the distribution of Weisfeiler-Lehman (WL)-embedded node vectors of a graph in a histogram that enables a dissimilarity measurement of two graphs using Wasserstein distance. It has been shown to produce better classification accuracy than other graph kernels which do not employ such distribution and Wasserstein distance. This paper introduces an alternative called Isolation Graph Kernel (IGK) that measures the similarity between two attributed graphs. IGK is unique in two aspects among existing graph kernels. First, it is the first graph kernel which employs a distributional kernel in the framework of kernel mean embedding. This avoids the need to use the computationally expensive Wasserstein distance. Second, it is the first graph kernel that incorporates the distribution of attributed nodes (ignoring the edges) in a dataset of graphs. We reveal that this distributional information, extracted in the form of a feature map of Isolation Kernel, is crucial in building an efficient and effective graph kernel. We show that IGK is better than WWL in terms of classification accuracy, and it runs orders of magnitude faster in large datasets when used in the context of SVM classification.

We introduce Isolation Distributional Kernel as a new way to measure the similarity between two distributions. Existing approaches based on kernel mean embedding, which converts a point kernel to a distributional kernel, have two key issues: the point kernel employed has a feature map with intractable dimensionality; and it is data independent. This paper shows that Isolation Distributional Kernel (IDK), which is based on a data dependent point kernel, addresses both key issues. We demonstrate IDK's efficacy and efficiency as a new tool for kernel based anomaly detection. Without explicit learning, using IDK alone outperforms existing kernel based anomaly detector OCSVM and other kernel mean embedding methods that rely on Gaussian kernel. We reveal for the first time that an effective kernel based anomaly detector based on kernel mean embedding must employ a characteristic kernel which is data dependent.

Set-level problems are as important as instance-level problems. The core in solving set-level problems is: how to measure the similarity between two sets. This paper investigates data-dependent kernels that are derived directly from data. We introduce Isolation Set-Kernel which is solely dependent on data distribution, requiring neither class information nor explicit learning. In contrast, most current set-similarities are not dependent on the underlying data distribution. We theoretically analyze the characteristic of Isolation Set-Kernel. As the set-kernel has a finite feature map, we show that it can be used to speed up the set-kernel computation significantly. We apply Isolation Set-Kernel to Multi-Instance Learning (MIL) using SVM classifier, and demonstrate that it outperforms other set-kernels or other solutions to the MIL problem.