Seminars & Colloquia
"Topological Signatures for Data Analysis Aided by Homological Generators "
Thursday March 03, 2022 10:15 AM
Location: 3211, EB2 NCSU Centennial Campus
Zoom Meeting Info (Visitor parking instructions)
Firstly, we investigate the problem of computing optimal representatives for persistent homology (a cornerstone in TDA). This work is motivated by the observation that traditional persistent homology only marks the birth and death of homological features without concrete representatives for the features. We thereby introduce the definition of persistent cycles as the representatives, which provide helpful visualization and reveal the geometry for the topological summaries. For computing optimal persistent cycles (providing the tightest representation), we prove the NP-hardness in general dimensions and propose algorithms for a special but useful class of inputs called manifolds.
Secondly, we look into a powerful extension of persistent homology called zigzag persistence, which enables shrinking of topological spaces besides growing and is especially useful when deletions of pieces are needed (e.g., a dynamical sequence of changing graphs). In this line of work, we propose near-linear algorithms for graphs, improving the previously known super-quadratic complexity. We also propose update algorithms for local changes on input filtrations, generating more advanced signatures called vines and vineyard.
As an application of the topological signatures, I will also discuss a collaborative work with researchers from Materials Science, where we devised a topological noise filter for microstructure segmentation for 3D images.
(These are joint work with my PhD advisor Tamal Dey.)
Host: Don Sheehy, CSC