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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.<\/jats:p>","DOI":"10.1007\/s10208-021-09522-y","type":"journal-article","created":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T20:02:30Z","timestamp":1626206550000},"page":"1069-1131","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["A Framework for Differential Calculus on Persistence Barcodes"],"prefix":"10.1007","volume":"22","author":[{"given":"Jacob","family":"Leygonie","sequence":"first","affiliation":[]},{"given":"Steve","family":"Oudot","sequence":"additional","affiliation":[]},{"given":"Ulrike","family":"Tillmann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,13]]},"reference":[{"issue":"1","key":"9522_CR1","first-page":"218","volume":"18","author":"Henry Adams","year":"2017","unstructured":"Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. 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