Approximating Continuous Functions on Persistence Diagrams Using Template Functions

Abstract

The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into vector spaces, in a way which maximizes the structure preserved. This process is commonly referred to as featurization. In this paper, we describe a mathematical framework for featurization called \emph{template functions}, and we show that it addresses the problem of approximating continuous functions on compact subsets of the space of persistence diagrams. Specifically, we begin by characterizing relative compactness with respect to the bottleneck distance, and then provide explicit theoretical methods for constructing compact-open dense subsets of continuous functions on persistence diagrams. These dense subsets -- obtained via template functions -- are leveraged for supervised learning tasks with persistence diagrams. Specifically, we test the method for classification and regression algorithms on several examples including shape data and dynamical systems.

To appear in Foundations of Computational Mathematics

References in corpus (6)

• Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems
• Sliced Wasserstein Kernel for Persistence Diagrams
• Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector
• Comparing persistence diagrams through complex vectors
• Adaptive Partitioning for Template Functions on Persistence Diagrams
• Persistence Flamelets: multiscale Persistent Homology for kernel density exploration

Cited by in corpus (9)

• Persistent Homology of Complex Networks for Dynamic State Detection
• A Survey of Vectorization Methods in Topological Data Analysis
• Topological Feature Vectors for Chatter Detection in Turning Processes
• Strong topology on the set of persistence diagrams
• Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector
• Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
• Adaptive Partitioning for Template Functions on Persistence Diagrams
• ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning
• Topological and metric properties of spaces of generalized persistence diagrams