A Framework for Differential Calculus on Persistence Barcodes

Abstract We define notions of differentiability for maps from and to the space persistence barcodes. Inspired by theory diffeological spaces, proposed framework uses lifts ordered barcodes, which derivatives can be computed. The two derived (respectively, barcodes) combine together naturally produce a chain rule that enables use gradient descent objective functions factoring through illustrate versatility this showing how it used analyze smoothness various parametrized families filtrations arising in topological data analysis.