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Topology"],"published-print":{"date-parts":[[2024,11]]},"abstract":"Abstract<\/jats:title>Let\u00a0K<\/jats:italic> be a finite simplicial, cubical, delta or CW complex. The persistence map\u00a0$$\\textrm{PH}$$<\/jats:tex-math>\n PH<\/mml:mtext>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> takes a filter\u00a0$$f:K\\rightarrow \\mathbb {R}$$<\/jats:tex-math>\n \n f<\/mml:mi>\n :<\/mml:mo>\n K<\/mml:mi>\n \u2192<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as input and returns the barcodes of the sublevel set persistent homology of\u00a0f<\/jats:italic> in each dimension. We address the inverse problem: given target barcodes\u00a0D<\/jats:italic>, computing the fiber\u00a0$$\\textrm{PH}^{-1}(D)$$<\/jats:tex-math>\n \n \n PH<\/mml:mtext>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n D<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For this, we use the fact that\u00a0$$\\textrm{PH}^{-1}(D)$$<\/jats:tex-math>\n \n \n PH<\/mml:mtext>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n D<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> decomposes as a polyhedral complex when\u00a0K<\/jats:italic> is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth-first search that recovers the polytopes forming the fiber\u00a0$$\\textrm{PH}^{-1}(D)$$<\/jats:tex-math>\n \n \n PH<\/mml:mtext>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n D<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.<\/jats:p>","DOI":"10.1007\/s41468-024-00165-w","type":"journal-article","created":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T18:02:08Z","timestamp":1713463328000},"page":"2015-2049","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Algorithmic reconstruction of the fiber of persistent homology on cell complexes"],"prefix":"10.1007","volume":"8","author":[{"given":"Jacob","family":"Leygonie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1889-8272","authenticated-orcid":false,"given":"Gregory","family":"Henselman-Petrusek","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,4,18]]},"reference":[{"issue":"1","key":"165_CR1","first-page":"218","volume":"18","author":"H Adams","year":"2017","unstructured":"Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: a stable vector representation of persistent homology. 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