A Canonical Framework for Summarizing Persistence Diagrams

Overview

Explore a canonical framework for summarizing persistence diagrams in this 59-minute lecture from the Applied Algebraic Topology Network. Delve into the Persistence Curve Framework, a novel approach to transforming persistence diagrams for compatibility with machine learning algorithms. Learn about the framework's definition, theoretical foundations including stability and stochastic convergence, and its applications in texture analysis, time series classification, and skin lesion analysis. Discover how this framework subsumes other summary functions like persistence landscapes and offers benefits for topological data analysis. Examine practical implementations through case studies on orbit classification, texture classification using KTH-TIPS2b dataset, skin lesion classification, and time series classification with various metrics and ensemble approaches.

Syllabus

Intro
Outline
Gray-scale images
Persistent Homology
The Space of Persistence Diagrams
Gaussian Persistence Curves
Benefits of Persistence Curves
Orbits Classification (Synthetic Data)
Data generation
Texture Classification: KTH-TIPS2b
Skin Lesion Classification: The Data
Skin Lesion Classification: The Model
Skin Lesion Classification: Results
Time Series Classification: The Data
Time Series Classification: Metrics
Time Series Classification: PC based distance?
Time Series Classification: Ensemble Metric
Time Series Classification: Results