Cheng Xin, Purdue University

Abstract

Topological Data Analysis (TDA) unfolds a rich tapestry of techniques rooted in algebraic topology, enabling a deep exploration of data's inherent shape and structure. Its versatility finds resonance across a spectrum of domains including computer science, biology, neuroscience, and physics, marking TDA as an inspiring approach to contemporary data analysis. This talk aims to demystify the foundational precepts of algebraic topology, segueing into a pivotal TDA tool - persistent homology, which characterizes the evolutionary trajectory of topological features through filtrations of topological spaces. Advancing further, we shall traverse the extension from 1-parameter persistent homology to multi-parameter persistence modules, spotlighting our innovative algorithm for decomposing persistence modules efficiently. An ensuing discussion will unveil a robust vector representation of 2-parameter persistence modules, pivoted on generalized rank invariants. Characterized by stability, 1-Lipschitz continuity, and differentiability almost everywhere, our representation emerges as a potent learnable entity within machine learning architectures like graph neural networks, potentially heralding a new frontier in topological data analytics.

See: https://theory.cs.rutgers.edu/theory_seminar