Anomaly detection on data streams presents significant challenges, requiring methods to maintain high detection accuracy among evolving distributions while ensuring real-time efficiency. Here we introduce IDK-S, a novel Incremental Distributional Kernel for Streaming anomaly detection that effectively addresses these challenges by creating a new dynamic representation in the kernel mean embedding framework. The superiority of IDK-S is attributed to two key innovations. First, it inherits the strengths of the Isolation Distributional Kernel, an offline detector that has demonstrated significant performance advantages over foundational methods like Isolation Forest and Local Outlier Factor due to the use of a data-dependent kernel. Second, it adopts a lightweight incremental update mechanism that significantly reduces computational overhead compared to the naive baseline strategy of performing a full model retraining. This is achieved without compromising detection accuracy, a claim supported by its statistical equivalence to the full retrained model. Our extensive experiments on thirteen benchmarks demonstrate that IDK-S achieves superior detection accuracy while operating substantially faster, in many cases by an order of magnitude, than existing state-of-the-art methods.

Topological Data Analysis (TDA) is used to extract topological features such as rings from point clouds. Recent works have identified that existing methods, which construct persistence diagrams in TDA, are not robust to noise and varied densities in a point cloud. This causes these methods to obtain incorrect topological features. We analyze the necessary properties of an approach that can address these two issues, and propose a new filter function for TDA based on a new data-dependent kernel that possesses these properties. Our empirical evaluation reveals that (i) the proposed kernel provides a better mean for UMAP dimensionality reduction (ii) the proposed filter function can significantly improve the performance of Topological Point Cloud Clustering (iii) the proposed filter function is a more effective way of constructing Persistence Diagram for t-SNE visualization and SVM classification than three existing methods of TDA, In addition, we explore the proposed filter’s performance on a more complex deformation named Riemannian stretching. Our proposed filter equipped with Sample Fermat distance outperforms all the other filters when noise and Riemannian stretching coexist. Code is available at https://github.com/IsolationKernel/Codes/tree/main/Lambda-kernel.

Recent works have identified that existing methods, which construct persistence diagrams in Topological Data Analysis (TDA), are not robust to noise and varied densities in a point cloud. We analyze the necessary properties of an approach that can address these two issues, and propose a new filter function for TDA based on a new data-dependent kernel which possesses these properties. Our empirical evaluation reveals that the proposed filter function provides a better means for t-SNE visualization and SVM classification than three existing methods of TDA.