Featurization of Topological Data Analysis Using Persistence Landscape and Circular Coordinates

• Activity Number: 542 - Advances in Topological and Geometric Data Analysis
• Type: Topic Contributed
• Date/Time: Thursday, August 11, 2022 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Jisu Kim* and Kwangho Kim and Manzil Zaheer and Joon Sik Kim and Frédéric Chazal and Larry Wasserman and Hengrui Luo and Alice Patania and Mikael Vejdemo-Johansson
• Companies: Inria and Harvard and Google Research and Carnegie Mellon University and Inria and Carnegie Mellon University and Lawrence Berkeley National Laboratory and Indiana University and CUNY
• Keywords: Topological Data Analysis; Persistent homology; Persistence landscape; Circular coordinates

Abstract

Topological Data Analysis generally refers to utilizing topological features from data. One of the main areas in topological data analysis is persistent homology, which observes data in different resolutions and summarizes topological features that persistently appear. However, directly applying persistent homology to statistical or machine learning frameworks is difficult due to its complex structure. To facilitate further application, the persistent homology is often featurized in Euclidean space or functional space. In this talk, I will explore how persistent homology can be featurized to be further applied in statistical or machine learning frameworks. Among featurizations for persistent homology, I will take a look at persistence landscape and circular coordinates. First, I will introduce persistence landscape and how it can be used to featurize time series data and build a topological layer. Then, I will introduce circular coordinates and how they can be used for visualization and dimension reduction.