A Framework for Differential Calculus on Persistence Barcodes

TL;DR

This paper introduces a framework for defining and computing derivatives of functions involving persistence barcodes, enabling gradient-based optimization in topological data analysis.

Contribution

It proposes a novel differentiability framework inspired by diffeological spaces, allowing derivatives to be computed via lifts to ordered barcodes.

Findings

Framework enables gradient descent on barcode-based functions

Analyzes smoothness of parametrized filtrations in topological data analysis

Provides a chain rule for derivatives involving barcodes

Abstract

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