A framework for differential calculus on persistence barcodes (Q2162119)

Leygonie et al. analyse the differentiability of so-called \textit{persistence barcodes} in this work. Persistence barcodes are topological descriptors that are arise when calculating topological features of filtered simplicial complexes using persistent homology. Originally introduced by \textit{H. Edelsbrunner} et al. [Discrete Comput. Geom. 28, No. 4, 511--533 (2002; Zbl 1011.68152)], persistent homology was considered to be a method for the extraction of \textit{static} topological features. With the rise of optimisation-based methods, in particular in machine learning, this perspective changed. A natural question in this context involves to what extent it is possible to calculate gradients of objective functions that factor through topological representations. In this work, Leygonie et al. show under which conditions there is a chain rule that enables the use of gradient descent algorithms, making it possible to differentiate objective functions that factor through the space of barcodes. As the main contribution, conditions are provided under which such maps are differentiable. This is followed by a discussion of relevant filtrations, viz. the lower-star and Rips filtrations, showing the requirements of filtrations functions to be differentiable in this setting. Moreover, the article provides a proof-of-concept discussion about when other types of topological representations, such as persistence images, are differentiable. Thus, the article answers highly relevant questions, opening a path towards more applications of persistent homology in the context of optimisation schemes.