Liz Munch - Featurization of Persistence Diagrams Using Template Functions for ML Tasks

Explore the mathematical framework for featurizing persistence diagrams using template functions in this 52-minute lecture from the Applied Algebraic Topology Network. Delve into the challenges of integrating persistence diagrams with machine learning techniques and discover how template functions can maximize preserved structure when mapping diagrams to Euclidean space. Learn about two exemplar template function families and their applications to synthetic and real datasets. Gain insights into topological data analysis, persistent homology, and the characterization of relatively compact sets. Examine various coordinate systems, including birth-lifetime coordinates, and understand how to combine functions and diagrams to derive numerical values. Investigate specific template systems such as tent functions and Chebychev polynomials, and explore experiments with random diagrams and manifolds. Conclude with an overview of current and future work in adaptive partitioning.

Intro
Shape in data
Topological Data Analysis (TDA)
Persistent homology in one slide
Existing Methods for Stats & ML
Finite vs infinte diagrams
A topologist's view of machine learning
Notation for persistence diagrams
8-matchings and Bottleneck Distance
Persistence diagram space is UGLY
Characterization of relatively compact sets
Up a creek?
Coordinate systems
Birth-Lifetime coordinates
Combining a function and a diagram to get a number
Evaluating points
Template function definition
Template functions
what about in practice?
Example template system 1: Tent functions
Example template system 2. Chebychev polynomials
Random diagrams
Manifold Experiment: Coefficients
Current and future work: Adaptive partitioning