Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms
This work addresses the problem of robust clustering in complex, high-dimensional datasets for data scientists, though it is incremental as it builds on existing spectral clustering methods with new metrics and guarantees.
The paper tackles clustering of elongated and irregularly shaped clusters in high-dimensional data with outliers, proving finite-sample guarantees for accuracy and correct cluster number determination using a longest-leg path distance metric and spectral clustering, with an efficient quasilinear runtime algorithm.
We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the labeling accuracy of the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient, easy to implement approximation algorithm for the LLPD based on a multiscale analysis of adjacency graphs, which allows for the runtime of LLPD spectral clustering to be quasilinear in the number of data points.