Theory of exact multi-parameter persistent homology and machine learning for extracting continuous features from discrete data Pohang University of Science and Technology

Abstract

In this dissertation, I propose a method called Exact Multi-parameter Persistent Homology (EMPH) for analyzing time-series data and its filtration learning. Typi- cally, data is provided discretely, so to analyze the topological properties of the data, one must consider a continuous transformation process known as filtration. Persis- tent homology is employed as a tool for tracking the changes in homology through- out the filtration. The outcome of topological data analysis varies depending on the choice of filtration, making the selection of an appropriate filtration critical. Filtration learning attempts to find an optimal filtration, which is a filtration that minimizes the loss function, using machine learning. EMPH is a method of topological time-series analysis that analyzes the Liouville torus of time-series data using multi-parameter persistent homology, under the assumption that the time-series data originates from an integrable Hamiltonian system. I deduce the rank invariant formula for a one- parameter filtration in the multi-parameter space, reveal the information encoded in the time-series data, and discuss how it can be applied to time-series analysis. Fil- tration learning in EMPH includes finding an optimal one-parameter filtration in the multi-parameter space, which can more accurately reflect the characteristics of the data and empirically demonstrates an enhancement in the classification accuracy of time-series data.