Persistence Terrace for Topological Inference of Point Cloud Data

• Activity Number: 293 - SPEED: Computing, Graphics, and Programming Statistics
• Type: Contributed
• Date/Time: Tuesday, August 1, 2017 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Graphics
• Author(s): Chul Moon* and Noah Giansiracusa and Nicole Lazar
• Companies: University of Georgia and Swarthmore College and University of Georgia
• Keywords: topological data analysis ; persistent homology ; density estimation ; visualization

Abstract

We propose a new plot, called the persistence terrace, that reveals the topological features of point cloud data. The topological characteristics of a topological space can be estimated robustly to noise and outliers by smoothing; construct a manifold with the point cloud using a smoothing function and compute persistent homology of the level set of the smoothed manifold. We construct manifolds with several smoothing parameters and apply persistent homology. The persistence terrace visualizes the computed persistent homology results of the manifolds. In the k-dimensional persistence terrace, we plot the kth Betti number against the filtration value and smoothing parameter. A k-dimensional hole is represented as a discrete staircase region; overlapping regions stack. The persistence terrace allows for the investigation of the topological characteristics of the space without the need to choose the optimal smoothing parameter. Furthermore, we can infer size and point density of a topological feature by examining the length of the corresponding region on the plot. In this talk, we will introduce the persistence terrace and demonstrate its ability to detect meaningful features in data.