Topological De-Noising: Strengthening the Topological Signal

Abstract: Topological methods, including persistent homology, are powerful tools for analysis of high-dimensional data sets but these methods rely almost exclusively on thresholding techniques. In noisy data sets, thresholding does not always allow for the recovery of topological information. We present an easy to implement, computationally efficient pre-processing algorithm to prepare noisy point cloud data sets for topological data analysis. The topological de-noising algorithm allows for the recovery of topological information that is inaccessible by thresholding methods. We apply the algorithm to synthetically-generated noisy data sets and show the recovery of topological information which is impossible to obtain by thresholding. We also apply the algorithm to natural image data in R^8 and show a very clean recovery of topological information previously only available with large amounts of thresholding. Finally, we discuss future directions for improving this algorithm using zig-zag persistence methods.