Persistent Homology on Streaming Data

Abstract

This paper introduces a framework to compute persistent homology, a principal tool in Topological Data Analysis, on potentially unbounded and evolving data streams. The framework is organized into online and offline components. The online element maintains a summary of the data that preserves the topological structure of the stream. The offline component computes the persistence intervals from the data captured by the summary. The framework is applied to the detection of horizontal or reticulate genomic exchanges during the evolution of species that cannot be identified by phylogenetic inference or traditional data mining. The method effectively detects reticulate evolution that occurs through reassortment and recombination in large streams of genomic sequences of Influenza and HIV viruses.