17:
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271:
40:
that captures density information about the underlying data set by filtering the points of the offsets at each index according to how many balls cover each point. The multicover bifiltration has been an object of study within multidimensional
151:
156:
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modules in all dimensions. However, the subdivision-ÄŚech bifiltration has an exponential number of simplices in the size of the data set, and hence is not amenable to efficient direct computations.
90:
293:
319:
363:
114:
389:
266:{\displaystyle \operatorname {Cov} _{r,k}:=\{b\in \mathbb {R} ^{d}:||b-a||\leq r{\text{ for at least }}k{\text{ points }}a\in A\}}
119:
796:
380:
The multicover bifiltration is also topologically equivalent to a multicover nerve construction due to Sheehy called the
46:
381:
60:
333:
The multicover bifiltration admits a topologically equivalent polytopal model of polynomial size, called the "
16:
687:
415:
Botnan, Magnus Bakke; Lesnick, Michael (2022). "An
Introduction to Multiparameter Persistence". p. 26.
385:
396:
345:
of the multicover bifiltration along one axis of the indexing set. The rhomboid bifiltration on a set of
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659:
342:
42:
388:
on the nerve of the offsets. In particular, the subdivision-ÄŚech and multicover bifiltrations are
276:
738:
632:
606:
579:
553:
505:
416:
33:
20:
The 2- and 3-fold covers of 7 points in the plane with respect to a particular scale parameter.
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29:
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599:"On the consistency and asymptotic normality of multiparameter persistent Betti numbers"
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Corbet, René; Kerber, Michael; Lesnick, Michael; Osang, Georg (2023-02-20).
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Corbet, René; Kerber, Michael; Lesnick, Michael; Osang, Georg (2023-02-20).
482:
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bifiltration." The rhomboid bifiltration is an extension of the rhomboid
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32:
derived from the covering of a finite set in a metric space by growing
652:"Multi-Parameter Persistent Homology is Practical (Extended Abstract)"
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558:
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421:
57:
Following the notation of Corbet et al. (2022), given a finite set
371:
15:
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introduced by
Edelsbrunner and Osang in 2021 for computing the
688:"Improvements to the Pipeline of Multiparameter Persistence"
146:{\displaystyle \mathbb {R} \times \mathbb {N} ^{\text{op}}}
376:
An example of the rhomboid tiling on a set of five points.
351:
301:
279:
159:
122:
102:
63:
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Blumberg, Andrew J.; Lesnick, Michael (2022-10-17).
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CCCG: Canadian conference in computational geometry
597:Botnan, Magnus B.; Hirsch, Christian (2022-12-22).
357:
313:
295:denotes the non-negative integers. Note that when
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265:
145:
108:
84:
441:"The Multi-Cover Persistence of Euclidean Balls"
546:"Stability of 2-Parameter Persistent Homology"
603:Journal of Applied and Computational Topology
8:
439:Edelsbrunner, Herbert; Osang, Georg (2021).
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179:
36:. It is a multidimensional extension of the
775:A multicover nerve for geometric inference
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116:is a two-parameter filtration indexed by
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85:{\displaystyle A\subset \mathbb {R} ^{d}}
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550:Foundations of Computational Mathematics
731:"Computing the Multicover Bifiltration"
498:"Computing the Multicover Bifiltration"
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28:is a two-parameter sequence of nested
735:Discrete & Computational Geometry
502:Discrete & Computational Geometry
445:Discrete & Computational Geometry
7:
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369:can be computed in polynomial time.
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650:Kerber, Michael (2022-07-29).
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1:
382:subdivision-ÄŚech bifiltration
288:{\displaystyle \mathbb {N} }
823:
753:10.1007/s00454-022-00476-8
621:10.1007/s41468-022-00110-9
568:10.1007/s10208-022-09576-6
520:10.1007/s00454-022-00476-8
457:10.1007/s00454-021-00281-9
47:topological data analysis
321:is fixed we recover the
240: for at least
386:barycentric subdivision
94:multicover bifiltration
26:multicover bifiltration
797:Computational geometry
703:Cite journal requires
667:Cite journal requires
384:, which considers the
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153:defined index-wise as
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686:Corbet, Rene (2020).
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343:persistent homology
314:{\displaystyle k=1}
43:persistent homology
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248: points
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30:topological spaces
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392:, and hence have
390:weakly equivalent
358:{\displaystyle n}
323:Offset Filtration
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109:{\displaystyle A}
38:offset filtration
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451:(4): 1296–1313.
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365:points in a
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34:metric balls
25:
23:
791:Categories
744:2103.07823
612:2109.05513
559:2010.09628
511:2103.07823
422:2203.14289
403:References
394:isomorphic
329:Properties
53:Definition
761:0179-5376
637:237491663
629:2367-1726
584:224705357
576:1615-3375
528:0179-5376
465:0179-5376
255:∈
233:≤
217:−
186:∈
129:×
68:⊂
807:Geometry
802:Topology
483:34720303
397:homology
335:rhomboid
273:, where
781:, 2012.
474:8550220
777:,” in
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635:
627:
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339:tiling
92:, the
739:arXiv
633:S2CID
607:arXiv
580:S2CID
554:arXiv
506:arXiv
417:arXiv
757:ISSN
709:help
673:help
625:ISSN
572:ISSN
524:ISSN
479:PMID
461:ISSN
45:and
24:The
749:doi
617:doi
564:doi
516:doi
469:PMC
453:doi
162:Cov
96:on
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353:n
309:1
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303:k
282:N
261:}
258:A
252:a
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229:|
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214:b
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191:R
183:b
180:{
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169:,
166:r
134:N
125:R
104:A
78:d
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65:A
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