Ulrich Bauer: Ripser - Efficient Computation of Vietoris–Rips Persistence Barcodes

Explore efficient computation of Vietoris–Rips persistence barcodes in this 51-minute lecture from the Hausdorff Trimester Program on Applied and Computational Algebraic Topology. Delve into Ripser, a compact C++ software for calculating persistence barcodes, and its design goals. Examine matrix reduction algorithms, persistent cohomology, and the fundamental theorem of discrete Morse theory. Learn about Morse pairs, persistence pairs, and apparent pairs, connecting Morse theory to persistence. Gain insights into Vietoris-Rips filtrations, compatible basis cycles, and oblivious matrix reduction techniques. Witness a live demonstration of Ripser's global reach, showcasing users from 156 different cities.

Intro
Vietoris-Rips filtrations
Demo: Ripser
Ripser A software for computing Vietoris-flips persistence barcodes - about 1000 lines of C++ code, no external dependencies
Design goals
The four special ingredients
Matrix reduction algorithm Setting: finite metric space X, points
Compatible basis cycles For a reduced boundary matrix R-D. V, call
Persistent cohomology
Counting cohomology column reductions
Observations
Implicit matrix reduction Standard approach
Oblivious matrix reduction
Natural filtration settings
Fundamental theorem of discrete Morse theory Letf be a discrete Morse function on a cell complex K.
Morse pairs and persistence pairs
Apparent pairs
From Morse theory to persistence and back
Ripser Live: users from 156 different cities