17:
983:
1160:
1195:
The offset filtration is also known to be stable with respect to perturbations of the underlying data set. This follows from the fact that the offset filtration can be viewed as a sublevel-set filtration with respect to the distance function of the metric space. The stability of sublevel-set
1191:
It is a well-known result of Niyogi, Smale, and
Weinberger that given a sufficiently dense random point cloud sample of a smooth submanifold in Euclidean space, the union of balls of a certain radius recovers the homology of the object via a deformation retraction of the Čech complex.
67:. 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
1566:-dimensional persistent homology barcode. While first stated in 2005, this sublevel stability result also follows directly from an algebraic stability property sometimes known as the "Isometry Theorem," which was proved in one direction in 2009, and the other direction in 2011.
1427:
885:
556:
1070:
1065:
615:
239:
353:
801:
803:
from the poset category of non-negative real numbers to the category of topological spaces and continuous maps. There are some advantages to the categorical viewpoint, as explored by
Bubenik and others.
648:
474:
880:
1322:
1539:
1186:
681:
2079:
Kim, Jisu; Shin, Jaehyeok; Chazal, Frédéric; Rinaldo, Alessandro; Wasserman, Larry (2020-05-12). "Homotopy
Reconstruction via the Cech Complex and the Vietoris-Rips Complex".
1317:
1220:
1499:
1564:
1291:
705:
413:
393:
263:
978:{\displaystyle \operatorname {Rips} _{\varepsilon }(X)\subset \operatorname {Cech} _{\varepsilon ^{\prime }}(X)\subset \operatorname {Rips} _{\varepsilon ^{\prime }}(X)}
1463:
1266:
159:
133:
1688:
1240:
1003:
733:
433:
373:
283:
101:
479:
71:. Homological persistence as introduced in these papers by Frosini and Robins was subsequently formalized by Edelsbrunner et al. in their seminal 2002 paper
1155:{\displaystyle \operatorname {Cech} _{\varepsilon }(X)\subset \operatorname {Rips} _{2\varepsilon }(X)\subset \operatorname {Cech} _{2\varepsilon }(X)}
847:
Although the
Vietoris-Rips filtration is not identical to the Čech filtration in general, it is an approximation in a sense. In particular, for a set
1876:
Bauer, Ulrich; Kerber, Michael; Roll, Fabian; Rolle, Alexander (2023-02-16). "A unified view on the functorial nerve theorem and its variations".
1188:, implying that the Rips and Cech filtrations are 2-interleaved with respect to the interleaving distance as introduced by Chazal et al. in 2009.
1008:
561:
164:
2310:
2262:
2111:
2055:
1847:
1664:
288:
833:
741:
59:. Utilizing a union of balls to approximate the shape of geometric objects was first suggested by Frosini in 1992 in the context of
2638:
2235:
Anai, Hirokazu; Chazal, Frédéric; Glisse, Marc; Ike, Yuichi; Inakoshi, Hiroya; Tinarrage, Raphaël; Umeda, Yuhei (2020-05-26).
620:
1569:
A multiparameter extension of the offset filtration defined by considering points covered by multiple balls is given by the
75:
Since then, the offset filtration has become a primary example in the study of computational topology and data analysis.
2633:
1824:, Lecture Notes in Computer Science, vol. 9294, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 705–716,
2643:
829:
441:
56:
1422:{\displaystyle d_{B}({\mathcal {B}}_{i}(\gamma ),{\mathcal {B}}_{i}(\kappa ))\leq d_{\infty }(\gamma ,\kappa )}
850:
708:
1703:
1570:
1504:
20:
The offset filtration at six scale parameters on a point cloud sampled from two circles of different sizes.
2648:
1574:
837:
2289:
Chazal, Frédéric; Cohen-Steiner, David; Glisse, Marc; Guibas, Leonidas J.; Oudot, Steve Y. (2009-06-08).
1165:
653:
16:
2137:"An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists"
1715:
1296:
1199:
1468:
824:
and the intersection of convex sets is convex. The nerve of the union of balls is also known as the
1816:
Halperin, Dan; Kerber, Michael; Shaharabani, Doron (2015), Bansal, Nikhil; Finocchi, Irene (eds.),
836:
to the Čech filtration (defined as the nerve of each offset across all scale parameters), so their
52:
1544:
1271:
690:
398:
378:
248:
2516:
2482:
2463:
2437:
2365:
2316:
2268:
2240:
2080:
2061:
2016:
1990:
1956:
1930:
1903:
1885:
1853:
1825:
1731:
1682:
242:
44:
2609:
2591:
2552:
2534:
2455:
2406:
2357:
2306:
2258:
2217:
2176:
2158:
2117:
2107:
2051:
2008:
1948:
1843:
1798:
1670:
1660:
1635:
1617:
1432:
684:
1245:
2599:
2583:
2542:
2526:
2447:
2396:
2347:
2298:
2250:
2207:
2166:
2148:
2043:
2000:
1940:
1895:
1835:
1788:
1750:
1723:
1625:
1607:
138:
106:
64:
825:
551:{\displaystyle {\mathcal {O}}(X):=\{X^{(\varepsilon )}\mid \varepsilon \in [0,\infty )\}}
2547:
2504:
2035:
1817:
1719:
2604:
2571:
2171:
2136:
1922:
1630:
1595:
1225:
988:
718:
418:
358:
268:
86:
2627:
2467:
2425:
2272:
2020:
1978:
1960:
1907:
1735:
817:
813:
2369:
2065:
2320:
1857:
2426:"The Theory of the Interleaving Distance on Multidimensional Persistence Modules"
2290:
1899:
1839:
841:
60:
2587:
2530:
2336:"Finding the Homology of Submanifolds with High Confidence from Random Samples"
1944:
2451:
2401:
2384:
2352:
2335:
2254:
2004:
1793:
1776:
1293:-dimensional homology modules on the sublevel-set filtrations with respect to
821:
2595:
2538:
2459:
2410:
2361:
2221:
2162:
2153:
2121:
2040:
Proceedings of the ninth annual symposium on
Computational geometry - SCG '93
2012:
1952:
1802:
1674:
1621:
1612:
2503:
Corbet, René; Kerber, Michael; Lesnick, Michael; Osang, Georg (2023-02-20).
2302:
2195:
68:
2613:
2556:
2212:
2180:
1639:
2295:
Proceedings of the twenty-fifth annual symposium on
Computational geometry
2101:
2047:
1654:
1196:
filtrations can be stated as follows: Given any two real-valued functions
2106:. J. Harer. Providence, R.I.: American Mathematical Society. p. 61.
738:
Note that it is also possible to view the offset filtration as a functor
48:
40:
1060:{\displaystyle \varepsilon ^{\prime }/\varepsilon \geq {\sqrt {2d/d+1}}}
610:{\displaystyle X^{(\varepsilon )}\subseteq X^{(\varepsilon ^{\prime })}}
234:{\displaystyle B(x,\varepsilon )=\{y\in X\mid d(x,y)\leq \varepsilon \}}
2042:. San Diego, California, United States: ACM Press. pp. 218–231.
1727:
1573:, and has also been an object of interest in persistent homology and
2521:
2245:
2085:
1935:
1890:
1708:
Intelligent Robots and
Computer Vision X: Algorithms and Techniques
348:{\textstyle X^{(\varepsilon )}:=\bigcup _{x\in X}B(x,\varepsilon )}
2442:
1995:
1830:
15:
2383:
Cohen-Steiner, David; Edelsbrunner, Herbert; Harer, John (2007).
51:
features of a data set. The offset filtration commonly arises in
1501:
denote the bottleneck and sup-norm distances, respectively, and
796:{\displaystyle {\mathcal {O}}(X):[0,\infty )\to \mathbf {Top} }
1596:"Topology Applied to Machine Learning: From Global to Local"
1511:
1368:
1342:
1017:
956:
924:
747:
659:
635:
597:
485:
2334:
Niyogi, Partha; Smale, Stephen; Weinberger, Shmuel (2008).
1702:
Frosini, Patrizio (1992-02-01). Casasent, David P. (ed.).
2483:"Lecture notes for AMAT 840: Multiparameter Persistence"
643:{\displaystyle \varepsilon \leq \varepsilon ^{\prime }}
291:
2291:"Proximity of persistence modules and their diagrams"
2196:"Coverage in sensor networks via persistent homology"
1656:
A short course in computational geometry and topology
1547:
1507:
1471:
1435:
1325:
1299:
1274:
1248:
1228:
1202:
1168:
1073:
1011:
991:
888:
853:
744:
721:
693:
656:
623:
564:
482:
444:
421:
401:
381:
361:
271:
251:
167:
141:
109:
89:
1921:
Blumberg, Andrew J.; Lesnick, Michael (2022-10-17).
1558:
1533:
1493:
1457:
1421:
1311:
1285:
1260:
1234:
1214:
1180:
1154:
1059:
997:
977:
874:
795:
727:
699:
675:
642:
609:
550:
468:
438:By considering the collection of offsets over all
427:
407:
387:
367:
347:
277:
257:
233:
153:
127:
95:
2572:"The Multi-Cover Persistence of Euclidean Balls"
1751:"Towards computing homology from approximations"
1923:"Stability of 2-Parameter Persistent Homology"
2103:Computational topology : an introduction
8:
2570:Edelsbrunner, Herbert; Osang, Georg (2021).
2194:de Silva, Vin; Ghrist, Robert (2007-04-25).
1777:"Topological Persistence and Simplification"
545:
502:
228:
189:
2135:Chazal, Frédéric; Michel, Bertrand (2021).
1977:Bubenik, Peter; Scott, Jonathan A. (2014).
1775:Edelsbrunner; Letscher; Zomorodian (2002).
1319:are point-wise finite dimensional, we have
816:shows that the union of balls has the same
469:{\displaystyle \varepsilon \in [0,\infty )}
73:Topological Persistence and Simplification.
1687:: CS1 maint: location missing publisher (
2603:
2546:
2520:
2441:
2400:
2351:
2297:. Aarhus Denmark: ACM. pp. 237–246.
2244:
2211:
2170:
2152:
2084:
1994:
1979:"Categorification of Persistent Homology"
1934:
1889:
1829:
1818:"The Offset Filtration of Convex Objects"
1792:
1629:
1611:
1551:
1546:
1516:
1510:
1509:
1506:
1476:
1470:
1440:
1434:
1398:
1373:
1367:
1366:
1347:
1341:
1340:
1330:
1324:
1298:
1278:
1273:
1247:
1227:
1201:
1167:
1131:
1103:
1078:
1072:
1067:. In general metric spaces, we have that
1041:
1033:
1022:
1016:
1010:
990:
955:
950:
923:
918:
893:
887:
875:{\displaystyle X\subset \mathbb {R} ^{d}}
866:
862:
861:
852:
782:
746:
745:
743:
720:
692:
658:
657:
655:
634:
622:
596:
588:
569:
563:
509:
484:
483:
481:
443:
420:
400:
380:
360:
315:
296:
290:
270:
250:
166:
140:
108:
88:
2430:Foundations of Computational Mathematics
1927:Foundations of Computational Mathematics
2505:"Computing the Multicover Bifiltration"
2036:"The union of balls and its dual shape"
1586:
985:between the Rips and Čech complexes on
1680:
2576:Discrete & Computational Geometry
2509:Discrete & Computational Geometry
2389:Discrete & Computational Geometry
2340:Discrete & Computational Geometry
2284:
2282:
1983:Discrete & Computational Geometry
1781:Discrete & Computational Geometry
1534:{\displaystyle {\mathcal {B}}_{i}(-)}
820:as its nerve, since closed balls are
47:used to detect the size and scale of
7:
2141:Frontiers in Artificial Intelligence
1972:
1970:
1704:"Measuring shapes by size functions"
1600:Frontiers in Artificial Intelligence
2385:"Stability of Persistence Diagrams"
1594:Adams, Henry; Moy, Michael (2021).
832:Therefore the offset filtration is
2200:Algebraic & Geometric Topology
1477:
1399:
773:
539:
460:
103:be a finite set in a metric space
14:
1181:{\displaystyle \varepsilon >0}
676:{\displaystyle {\mathcal {O}}(X)}
789:
786:
783:
2239:. Abel Symposia. Vol. 15.
1312:{\displaystyle \gamma ,\kappa }
1215:{\displaystyle \gamma ,\kappa }
828:, which is a subcomplex of the
2100:Edelsbrunner, Herbert (2010).
2034:Edelsbrunner, Herbert (1993).
1749:Robins, Vanessa (1999-01-01).
1653:Edelsbrunner, Herbert (2014).
1528:
1522:
1494:{\displaystyle d_{\infty }(-)}
1488:
1482:
1452:
1446:
1416:
1404:
1388:
1385:
1379:
1359:
1353:
1336:
1149:
1143:
1121:
1115:
1093:
1087:
972:
966:
940:
934:
908:
902:
882:we have a chain of inclusions
812:A standard application of the
779:
776:
764:
758:
752:
670:
664:
602:
589:
576:
570:
542:
530:
516:
510:
496:
490:
463:
451:
375:with respect to the parameter
342:
330:
303:
297:
219:
207:
183:
171:
122:
110:
1:
1900:10.1016/j.exmath.2023.04.005
1840:10.1007/978-3-662-48350-3_59
1559:{\displaystyle i{\text{th}}}
1286:{\displaystyle i{\text{th}}}
700:{\displaystyle \varepsilon }
408:{\displaystyle \varepsilon }
388:{\displaystyle \varepsilon }
258:{\displaystyle \varepsilon }
2665:
2588:10.1007/s00454-021-00281-9
2531:10.1007/s00454-022-00476-8
2490:University at Albany, SUNY
1945:10.1007/s10208-022-09576-6
476:we get a family of spaces
355:is known as the offset of
2481:Lesnick, Michael (2023).
2452:10.1007/s10208-015-9255-y
2424:Lesnick, Michael (2015).
2402:10.1007/s00454-006-1276-5
2353:10.1007/s00454-008-9053-2
2255:10.1007/978-3-030-43408-3
2237:Topological Data Analysis
2005:10.1007/s00454-014-9573-x
1878:Expositiones Mathematicae
1794:10.1007/s00454-002-2885-2
57:topological data analysis
2154:10.3389/frai.2021.667963
1613:10.3389/frai.2021.668302
1458:{\displaystyle d_{B}(-)}
2303:10.1145/1542362.1542407
1714:. Boston, MA: 122–133.
1571:multicover bifiltration
1261:{\displaystyle i\geq 0}
1222:on a topological space
2639:Computational topology
2213:10.2140/agt.2007.7.339
1575:computational geometry
1560:
1535:
1495:
1459:
1423:
1313:
1287:
1262:
1236:
1216:
1182:
1156:
1061:
999:
979:
876:
830:Vietoris-Rips complex.
797:
729:
701:
683:is a family of nested
677:
644:
611:
552:
470:
429:
409:
389:
369:
349:
279:
259:
235:
155:
154:{\displaystyle x\in X}
129:
97:
21:
2048:10.1145/160985.161139
1822:Algorithms - ESA 2015
1561:
1536:
1496:
1460:
1424:
1314:
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678:
645:
612:
553:
471:
430:
410:
390:
370:
350:
280:
260:
236:
156:
130:
128:{\displaystyle (M,d)}
98:
19:
1758:Topology Proceedings
1659:. Cham. p. 35.
1545:
1505:
1469:
1433:
1323:
1297:
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1246:
1226:
1200:
1166:
1071:
1009:
989:
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742:
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562:
480:
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379:
359:
289:
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249:
165:
139:
107:
87:
2634:Applied mathematics
1720:1992SPIE.1607..122F
53:persistent homology
2644:Geometric topology
1556:
1531:
1491:
1455:
1419:
1309:
1283:
1258:
1242:such that for all
1232:
1212:
1178:
1152:
1057:
995:
975:
872:
793:
725:
707:, which defines a
697:
685:topological spaces
673:
640:
607:
548:
466:
425:
405:
385:
365:
345:
326:
275:
255:
231:
151:
125:
93:
22:
2312:978-1-60558-501-7
2264:978-3-030-43407-6
2113:978-0-8218-4925-5
2057:978-0-89791-582-3
1849:978-3-662-48349-7
1666:978-3-319-05957-0
1554:
1281:
1235:{\displaystyle T}
1055:
998:{\displaystyle X}
834:weakly equivalent
728:{\displaystyle X}
713:offset filtration
428:{\displaystyle X}
368:{\displaystyle X}
311:
285:. Then the union
278:{\displaystyle x}
96:{\displaystyle X}
55:and the field of
28:(also called the
26:offset filtration
2656:
2618:
2617:
2607:
2582:(4): 1296–1313.
2567:
2561:
2560:
2550:
2524:
2500:
2494:
2493:
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2478:
2472:
2471:
2445:
2421:
2415:
2414:
2404:
2380:
2374:
2373:
2355:
2346:(1–3): 419–441.
2331:
2325:
2324:
2286:
2277:
2276:
2248:
2232:
2226:
2225:
2215:
2191:
2185:
2184:
2174:
2156:
2132:
2126:
2125:
2097:
2091:
2090:
2088:
2076:
2070:
2069:
2031:
2025:
2024:
1998:
1974:
1965:
1964:
1938:
1918:
1912:
1911:
1893:
1873:
1867:
1866:
1865:
1864:
1833:
1813:
1807:
1806:
1796:
1772:
1766:
1765:
1755:
1746:
1740:
1739:
1728:10.1117/12.57059
1699:
1693:
1692:
1686:
1678:
1650:
1644:
1643:
1633:
1615:
1591:
1565:
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1402:
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1110:
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232:
160:
158:
157:
152:
134:
132:
131:
126:
102:
100:
99:
94:
34:"union-of-disks"
30:"union-of-balls"
2664:
2663:
2659:
2658:
2657:
2655:
2654:
2653:
2624:
2623:
2622:
2621:
2569:
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2564:
2502:
2501:
2497:
2485:
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2423:
2422:
2418:
2382:
2381:
2377:
2333:
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2328:
2313:
2288:
2287:
2280:
2265:
2234:
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2229:
2193:
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2188:
2134:
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2129:
2114:
2099:
2098:
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2078:
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2073:
2058:
2033:
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2028:
1976:
1975:
1968:
1920:
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1915:
1875:
1874:
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1815:
1814:
1810:
1774:
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1753:
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1743:
1701:
1700:
1696:
1679:
1667:
1652:
1651:
1647:
1593:
1592:
1588:
1583:
1543:
1542:
1508:
1503:
1502:
1472:
1467:
1466:
1436:
1431:
1430:
1394:
1365:
1339:
1326:
1321:
1320:
1295:
1294:
1270:
1269:
1244:
1243:
1224:
1223:
1198:
1197:
1164:
1163:
1127:
1099:
1074:
1069:
1068:
1012:
1007:
1006:
987:
986:
951:
946:
919:
914:
889:
884:
883:
860:
849:
848:
838:homology groups
810:
740:
739:
717:
716:
689:
688:
652:
651:
630:
619:
618:
592:
584:
565:
560:
559:
505:
478:
477:
440:
439:
417:
416:
397:
396:
395:(or simply the
377:
376:
357:
356:
292:
287:
286:
267:
266:
247:
246:
163:
162:
137:
136:
105:
104:
85:
84:
81:
65:Euclidean space
39:) is a growing
12:
11:
5:
2662:
2660:
2652:
2651:
2646:
2641:
2636:
2626:
2625:
2620:
2619:
2562:
2515:(2): 376–405.
2495:
2473:
2436:(3): 613–650.
2416:
2395:(1): 103–120.
2375:
2326:
2311:
2278:
2263:
2227:
2206:(1): 339–358.
2186:
2127:
2112:
2092:
2071:
2056:
2026:
1989:(3): 600–627.
1966:
1913:
1868:
1848:
1808:
1787:(4): 511–533.
1767:
1741:
1694:
1665:
1645:
1585:
1584:
1582:
1579:
1550:
1530:
1527:
1524:
1519:
1513:
1490:
1487:
1484:
1479:
1475:
1454:
1451:
1448:
1443:
1439:
1418:
1415:
1412:
1409:
1406:
1401:
1397:
1393:
1390:
1387:
1384:
1381:
1376:
1370:
1364:
1361:
1358:
1355:
1350:
1344:
1338:
1333:
1329:
1308:
1305:
1302:
1277:
1257:
1254:
1251:
1231:
1211:
1208:
1205:
1177:
1174:
1171:
1151:
1148:
1145:
1142:
1137:
1134:
1130:
1126:
1123:
1120:
1117:
1114:
1109:
1106:
1102:
1098:
1095:
1092:
1089:
1086:
1081:
1077:
1054:
1051:
1048:
1044:
1040:
1037:
1032:
1029:
1025:
1019:
1015:
994:
974:
971:
968:
965:
958:
954:
949:
945:
942:
939:
936:
933:
926:
922:
917:
913:
910:
907:
904:
901:
896:
892:
869:
864:
859:
856:
809:
806:
791:
788:
785:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
749:
724:
696:
672:
669:
666:
661:
637:
633:
629:
626:
604:
599:
595:
591:
587:
583:
578:
575:
572:
568:
547:
544:
541:
538:
535:
532:
529:
526:
523:
518:
515:
512:
508:
504:
501:
498:
495:
492:
487:
465:
462:
459:
456:
453:
450:
447:
424:
404:
384:
364:
344:
341:
338:
335:
332:
329:
324:
321:
318:
314:
310:
305:
302:
299:
295:
274:
254:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
170:
150:
147:
144:
135:, and for any
124:
121:
118:
115:
112:
92:
80:
77:
13:
10:
9:
6:
4:
3:
2:
2661:
2650:
2649:Data analysis
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2631:
2629:
2615:
2611:
2606:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2566:
2563:
2558:
2554:
2549:
2544:
2540:
2536:
2532:
2528:
2523:
2518:
2514:
2510:
2506:
2499:
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2484:
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2469:
2465:
2461:
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2435:
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2420:
2417:
2412:
2408:
2403:
2398:
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2390:
2386:
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2363:
2359:
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2341:
2337:
2330:
2327:
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2318:
2314:
2308:
2304:
2300:
2296:
2292:
2285:
2283:
2279:
2274:
2270:
2266:
2260:
2256:
2252:
2247:
2242:
2238:
2231:
2228:
2223:
2219:
2214:
2209:
2205:
2201:
2197:
2190:
2187:
2182:
2178:
2173:
2168:
2164:
2160:
2155:
2150:
2146:
2142:
2138:
2131:
2128:
2123:
2119:
2115:
2109:
2105:
2104:
2096:
2093:
2087:
2082:
2075:
2072:
2067:
2063:
2059:
2053:
2049:
2045:
2041:
2037:
2030:
2027:
2022:
2018:
2014:
2010:
2006:
2002:
1997:
1992:
1988:
1984:
1980:
1973:
1971:
1967:
1962:
1958:
1954:
1950:
1946:
1942:
1937:
1932:
1928:
1924:
1917:
1914:
1909:
1905:
1901:
1897:
1892:
1887:
1883:
1879:
1872:
1869:
1859:
1855:
1851:
1845:
1841:
1837:
1832:
1827:
1823:
1819:
1812:
1809:
1804:
1800:
1795:
1790:
1786:
1782:
1778:
1771:
1768:
1763:
1759:
1752:
1745:
1742:
1737:
1733:
1729:
1725:
1721:
1717:
1713:
1709:
1705:
1698:
1695:
1690:
1684:
1676:
1672:
1668:
1662:
1658:
1657:
1649:
1646:
1641:
1637:
1632:
1627:
1623:
1619:
1614:
1609:
1605:
1601:
1597:
1590:
1587:
1580:
1578:
1576:
1572:
1567:
1548:
1525:
1517:
1485:
1473:
1449:
1441:
1437:
1413:
1410:
1407:
1395:
1391:
1382:
1374:
1362:
1356:
1348:
1331:
1327:
1306:
1303:
1300:
1275:
1255:
1252:
1249:
1229:
1209:
1206:
1203:
1193:
1189:
1175:
1172:
1169:
1146:
1140:
1135:
1132:
1128:
1124:
1118:
1112:
1107:
1104:
1100:
1096:
1090:
1084:
1079:
1075:
1052:
1049:
1046:
1042:
1038:
1035:
1030:
1027:
1023:
1013:
992:
969:
963:
952:
947:
943:
937:
931:
920:
915:
911:
905:
899:
894:
890:
867:
857:
854:
845:
843:
839:
835:
831:
827:
823:
819:
818:homotopy type
815:
814:nerve theorem
807:
805:
770:
767:
761:
755:
736:
722:
714:
711:known as the
710:
694:
687:indexed over
686:
667:
631:
627:
624:
593:
585:
581:
573:
566:
536:
533:
527:
524:
521:
513:
506:
499:
493:
457:
454:
448:
445:
436:
422:
402:
382:
362:
339:
336:
333:
327:
322:
319:
316:
312:
308:
300:
293:
272:
252:
244:
225:
222:
216:
213:
210:
204:
201:
198:
195:
192:
186:
180:
177:
174:
168:
148:
145:
142:
119:
116:
113:
90:
78:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
35:
31:
27:
18:
2579:
2575:
2565:
2512:
2508:
2498:
2489:
2476:
2433:
2429:
2419:
2392:
2388:
2378:
2343:
2339:
2329:
2294:
2236:
2230:
2203:
2199:
2189:
2144:
2140:
2130:
2102:
2095:
2074:
2039:
2029:
1986:
1982:
1926:
1916:
1881:
1877:
1871:
1861:, retrieved
1821:
1811:
1784:
1780:
1770:
1761:
1757:
1744:
1711:
1707:
1697:
1655:
1648:
1603:
1599:
1589:
1568:
1541:denotes the
1194:
1190:
846:
826:Čech complex
811:
737:
712:
437:
265:centered at
82:
72:
61:submanifolds
45:metric balls
36:
33:
29:
25:
23:
415:-offset of
243:closed ball
49:topological
2628:Categories
2522:2103.07823
2246:1811.04757
2147:: 667963.
2086:1903.06955
1936:2010.09628
1891:2203.03571
1863:2023-02-25
1764:: 503–532.
1581:References
842:isomorphic
808:Properties
709:filtration
245:of radius
79:Definition
69:attractors
37:filtration
2596:0179-5376
2539:0179-5376
2468:254158297
2460:1615-3375
2443:1106.5305
2411:0179-5376
2362:0179-5376
2273:242491854
2222:1472-2739
2163:2624-8212
2122:427757156
2021:254027425
2013:0179-5376
1996:1205.3669
1961:224705357
1953:1615-3375
1908:247291819
1831:1407.6132
1803:0179-5376
1736:121295508
1683:cite book
1675:879343648
1622:2624-8212
1526:−
1486:−
1478:∞
1450:−
1414:κ
1408:γ
1400:∞
1392:≤
1383:κ
1357:γ
1307:κ
1301:γ
1253:≥
1210:κ
1204:γ
1170:ε
1141:
1136:ε
1125:⊂
1113:
1108:ε
1097:⊂
1085:
1080:ε
1031:≥
1028:ε
1018:′
1014:ε
1005:whenever
964:
957:′
953:ε
944:⊂
932:
925:′
921:ε
912:⊂
900:
895:ε
858:⊂
780:→
774:∞
695:ε
636:′
632:ε
628:≤
625:ε
617:whenever
598:′
594:ε
582:⊆
574:ε
540:∞
528:∈
525:ε
522:∣
514:ε
461:∞
449:∈
446:ε
403:ε
383:ε
340:ε
320:∈
313:⋃
301:ε
253:ε
226:ε
223:≤
202:∣
196:∈
181:ε
146:∈
2614:34720303
2557:37581017
2548:10423148
2181:34661095
1884:(4): 8.
1640:34056580
1162:for all
41:sequence
2605:8550220
2370:1788129
2172:8511823
2066:9599628
1716:Bibcode
1631:8160457
241:be the
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1734:
1673:
1663:
1638:
1628:
1620:
1429:where
1268:, the
822:convex
558:where
2517:arXiv
2486:(PDF)
2464:S2CID
2438:arXiv
2366:S2CID
2317:S2CID
2269:S2CID
2241:arXiv
2081:arXiv
2062:S2CID
2017:S2CID
1991:arXiv
1957:S2CID
1931:arXiv
1904:S2CID
1886:arXiv
1854:S2CID
1826:arXiv
1754:(PDF)
1732:S2CID
1606:: 2.
650:. So
2610:PMID
2592:ISSN
2553:PMID
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2456:ISSN
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2358:ISSN
2307:ISBN
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2218:ISSN
2177:PMID
2159:ISSN
2118:OCLC
2108:ISBN
2052:ISBN
2009:ISSN
1949:ISSN
1844:ISBN
1799:ISSN
1712:1607
1689:link
1671:OCLC
1661:ISBN
1636:PMID
1618:ISSN
1465:and
1173:>
1129:Cech
1101:Rips
1076:Cech
948:Rips
916:Cech
891:Rips
840:are
161:let
83:Let
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