Differential Calculus on Persistence Barcodes

Explore differential calculus on persistence barcodes in this 29-minute conference talk by Jacob Leygonie. Delve into the definition of differentiability for maps from and to the space of persistence barcodes, inspired by the theory of diffeological spaces. Learn about the framework that utilizes lifts to the space of ordered barcodes for computing derivatives. Discover how the two derived notions of differentiability combine to produce a chain rule, enabling gradient descent for objective functions factoring through the space of barcodes. Examine the versatility of this framework through its application in analyzing the smoothness of various parametrized families of filtrations in topological data analysis. The talk is based on a preprint available at https://arxiv.org/pdf/1910.00960.pdf and is presented as part of the Applied Algebraic Topology Network.