Cluster Identification via Persistent Homology and Other Clustering Techniques, with Application to Liver Transplant Data

Abstract

Clustering, an unsupervised learning method, can be very useful in detecting hidden patterns in complex and/or high-dimensional data. Persistent homology, a recently developed branch of computational topology, studies the evolution of topological features under a varying filtration parameter. At a fixed filtration parameter value, one can find different topological features in a dataset, such as connected components (zero-dimensional topological features), loops (one-dimensional topological features), and more generally, k-dimensional holes (k-dimensional topological features). In the classical sense, clusters correspond to zero-dimensional topological features. We explore whether higher dimensional homology can contribute to detecting hidden patterns in data. We observe that some loops formed in survival data seem to be able to detect outliers that other clustering techniques do not detect. We analyze patterns of patients in terms of their covariates and survival time, and determine the most important predictor variables in predicting survival times of liver transplant patients by applying a random survival forest.