  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.
