Polynomial-Sized Topological Approximations Using the Permutahedron

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dc.contributor.authorChoudhary, Arunien
dc.contributor.authorKerber, Michaelen
dc.contributor.authorRaghvendra, Sharathen
dc.contributor.departmentComputer Scienceen
dc.date.accessioned2019-08-30T17:00:07Zen
dc.date.available2019-08-30T17:00:07Zen
dc.date.issued2019-01en
dc.description.abstractClassical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in Rd, we obtain a O(d)-approximation whose k-skeleton has size n2O(dlogk) per scale and n2O(dlogd) in total over all scales. In conjunction with dimension reduction techniques, our approach yields a O(polylog(n))-approximation of size nO(1) for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximation: we construct a point set for which every (1+epsilon)-approximation of the ech filtration has to contain n(loglogn) features, provided that epsilon < 1log1+cn for c(0,1).en
dc.description.notesOpen access funding provided by the Max Planck Society. Sharath Raghvendra acknowledges support of NSF CRII Grant CCF-1464276. Michael Kerber is supported by the Austrian Science Fund (FWF) grant number P 29984-N35.en
dc.description.sponsorshipMax Planck Society; NSF CRII [CCF-1464276]; Austrian Science Fund (FWF) [P 29984-N35]en
dc.format.mimetypeapplication/pdfen
dc.identifier.doihttps://doi.org/10.1007/s00454-017-9951-2en
dc.identifier.eissn1432-0444en
dc.identifier.issn0179-5376en
dc.identifier.issue1en
dc.identifier.urihttp://hdl.handle.net/10919/93322en
dc.identifier.volume61en
dc.language.isoenen
dc.rightsCreative Commons Attribution 4.0 Internationalen
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/en
dc.subjectPersistent homologyen
dc.subjectTopological data analysisen
dc.subjectSimplicial approximationen
dc.subjectPermutahedronen
dc.subjectApproximation algorithmsen
dc.subject55U10en
dc.subject11H06en
dc.subject68W25en
dc.titlePolynomial-Sized Topological Approximations Using the Permutahedronen
dc.title.serialDiscrete & Computational Geometryen
dc.typeArticle - Refereeden
dc.type.dcmitypeTexten
dc.type.dcmitypeStillImageen

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