Offset filtration: Difference between revisions
←Created page with 'The offset filtration (also called the "union-of-balls"<ref>{{Cite journal |last=Adams |first=Henry |last2=Moy |first2=Michael |date=2021 |title=Topology Applied to Machine Learning: From Global to Local |url=https://www.frontiersin.org/articles/10.3389/frai.2021.668302 |journal=Frontiers in Artificial Intelligence |volume=4 |pages=2 |doi=10.3389/frai.2021.668302 |issn=2624-8212 |pmc=PMC8160457 |pmid=34056580}}</ref> or "union-of-disks"<ref>{{Cite book |las...' |
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Revision as of 08:39, 25 February 2023
The offset filtration (also called the "union-of-balls"[1] or "union-of-disks"[2] filtration) is a growing sequence of metric balls used to detect the size and scale of topological features of a data set. The offset filtration commonly arises in persistent homology and the field of topological data analysis. Utilizing a union of balls to approximate the shape of geometric objects was first suggested by Frosini in 1992 in the context of submanifolds of Euclidean space.[3] The construction was independently explored by Robins in 1998, and expanded to considering the collection of offsets indexed over a series of increasing scale parameters (i.e., a growing sequence of balls), in order to observe the stability of topological features with respect to attractors.[4] Homological persistence as introduced in these papers by Frosini and Robins was subsequently formalized by Edelsbrunner et al. in their seminal 2002 paper Topological Persistence and Simplification.[5] Since then, the offset filtration has become a primary example in the study of computational topology and data analysis.
Definition
- ^ Adams, Henry; Moy, Michael (2021). "Topology Applied to Machine Learning: From Global to Local". Frontiers in Artificial Intelligence. 4: 2. doi:10.3389/frai.2021.668302. ISSN 2624-8212. PMC 8160457. PMID 34056580.
{{cite journal}}: CS1 maint: PMC format (link) CS1 maint: unflagged free DOI (link) - ^ Edelsbrunner, Herbert (2014). A short course in computational geometry and topology. Cham. p. 35. ISBN 978-3-319-05957-0. OCLC 879343648.
{{cite book}}: CS1 maint: location missing publisher (link) - ^ Frosini, Patrizio (1992-02-01). Casasent, David P. (ed.). "Measuring shapes by size functions". Boston, MA: 122–133. doi:10.1117/12.57059.
{{cite journal}}: Cite journal requires|journal=(help) - ^ Robins, Vanessa (1999-01-01). "Towards computing homology from approximations" (PDF). Topology Proceedings. 24: 503–532.
- ^ Edelsbrunner; Letscher; Zomorodian (2002). "Topological Persistence and Simplification". Discrete & Computational Geometry. 28 (4): 511–533. doi:10.1007/s00454-002-2885-2. ISSN 0179-5376.