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256:. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. Any attempt to include those points on the opposite side will necessarily include other points not in that pair. Hence, any pair of opposite points cannot be isolated out of
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956:, then S_C(3) would only be 7. If none of those sets can be obtained, S_C(3) would be 0. Additionally, if S_C(2)=3, for example, then there is an element in the set of all 2-point sets from
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gets smaller, there are fewer sets that could be omitted. The extreme of this is S_C(0) (the shattering coefficient of the empty set), which must always be
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656:{\displaystyle S_{C}(n)=\max _{\forall x_{1},x_{2},\dots ,x_{n}\in \Omega }\operatorname {card} \{\,\{\,x_{1},x_{2},\dots ,x_{n}\}\cap s,s\in C\}}
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With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all
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1777:. The VC dimension is usually defined as VC_0, the largest cardinality of points chosen that will still shatter
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A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using
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to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set
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gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of
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because, in empirical processes, we are interested in the shattering of finite sets of data points.
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Wencour, R. S.; Dudley, R. M. (1981), "Some special VapnikâChervonenkis classes",
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of four particular points on the unit circle (the unit circle is shown in purple).
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Each subset of adjacent points can be isolated with a disc (showing one of four).
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31:, also known as VC-theory. Shattering and VC-theory are used in the study of
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206:(two-dimensional space) does not shatter every set of four points on the
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1704:{\displaystyle VC_{0}(C)={\underset {n}{\max }}\{n:S_{C}(n)=2^{n}\}.\,}
1598:{\displaystyle VC(C)={\underset {n}{\min }}\{n:S_{C}(n)<2^{n}\}\,}
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Each individual point can be isolated with a disc (showing all four).
964:. It follows from this that S_C(3) would also be less than 8 (i.e.
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890:, its shattering coefficient(3) would be 8 and S_C(2) would be
252:, we attempt to draw a disc around every subset of points in
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A subset of points on opposite sides of the unit circle can
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in the plane does shatter every finite set of points on the
1984:, Ph.D. thesis, Stanford University, Mathematics Department
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is also separable by some ellipse (showing one of four)
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then it can not shatter sets of larger cardinalities.
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are now separable by some ellipse (showing one of two)
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1994:"Empirical discrepancies and subadditive processes"
748:{\displaystyle x_{1},x_{2},\dots ,x_{n}\in \Omega }
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1909:A class with finite VC dimension is called a
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1982:Combinatorial Entropy and Uniform Limit Laws
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975:This example illustrates some properties of
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2036:Origin of "Shattered sets" terminology
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679:{\displaystyle \operatorname {card} }
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923:. However, if one of those sets in
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198:We will show that the class of all
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1147:{\displaystyle A\subseteq \Omega }
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1056:{\displaystyle S_{C}(n)\leq 2^{n}}
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1923:uniformly Glivenko–Cantelli
1371:The third property means that if
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1259:{\displaystyle S_{C}(N)<2^{N}}
392:(visualize connecting the dots).
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441:{\displaystyle s\subset \Omega }
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29:Vapnik–Chervonenkis theory
16:Notion in computational learning
1911:Vapnik–Chervonenkis class
1383:Vapnik–Chervonenkis class
410:of sets, we use the concept of
379:Each subset of three points in
260:using intersections with class
1891:{\displaystyle S_{C}(n)=2^{n}}
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1969:10.1016/0012-365X(81)90274-0
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809:with the sets in collection
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324:be isolated by any disc in
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468:being any space, often a
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121:{\displaystyle a=c\cap A.}
35:as well as in statistical
1004:{\displaystyle S_{C}(n)}
790:{\displaystyle S_{C}(n)}
343:However, if we redefine
328:, we conclude then that
315:be isolated with a disc.
82:, there is some element
1497:{\displaystyle 2^{0}=1}
1397:cannot be shattered by
1360:{\displaystyle n\geq N}
916:{\displaystyle 2^{2}=4}
875:{\displaystyle 2^{3}=8}
461:{\displaystyle \Omega }
412:shattering coefficients
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2013:10.1214/aop/1176995615
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416:growth function
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2038:, by J. Steele
2031:
2030:External links
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2006:(1): 118â227,
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1962:(3): 313â318,
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610:
605:
601:
597:
592:
588:
576:
573:
565:
560:
556:
552:
549:
546:
541:
537:
533:
528:
524:
512:
506:
498:
494:
486:
485:
484:
482:
478:
475:
471:
432:
429:
421:
417:
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409:
403:
395:
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382:
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370:
366:
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346:
341:
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335:
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323:
314:
307:
302:
295:
290:
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267:
263:
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251:
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239:
234:
230:
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224:
219:
217:
213:
209:
205:
201:
193:
191:
189:
185:
181:
176:
174:
170:
166:
162:
158:
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150:
146:
142:
138:
134:
115:
112:
109:
106:
103:
100:
93:
92:
91:
89:
85:
81:
77:
73:
69:
66:
62:
58:
54:
50:
42:
40:
38:
34:
30:
26:
22:
2003:
1997:
1981:
1959:
1953:
1918:
1914:
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1908:
1903:
1899:
1846:
1842:
1838:
1836:
1782:
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1713:
1607:
1509:
1506:VC dimension
1468:
1437:
1433:
1402:
1398:
1394:
1392:
1389:VC dimension
1376:
1372:
1370:
1211:
1207:
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969:
965:
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761:
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686:denotes the
665:
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470:sample space
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143:is equal to
141:intersection
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21:intersection
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1508:of a class
1432:because as
849:, contains
759:points,.
688:cardinality
390:convex sets
216:unit circle
212:convex sets
208:unit circle
139:when their
2046:Categories
1946:References
1917:. A class
1785:such that
1714:Note that
90:such that
43:Definition
1352:≥
1266:for some
1142:Ω
1139:⊆
1104:⊆
1095:∈
1081:∩
1041:≤
886:shatters
743:Ω
740:∈
724:…
645:∈
633:∩
614:…
577:
569:Ω
566:∈
550:…
521:∀
456:Ω
436:Ω
433:⊂
248:shatters
149:power set
135:shatters
110:∩
1992:(1978),
1980:(1975),
1929:See also
1915:VC class
1898:for all
1341:for all
1128:for any
1067:because
1063:for all
422:of sets
236:The set
70:the set
68:shatters
47:Suppose
2022:2242865
1849:, then
264:and so
202:in the
194:Example
2020:
1781:(i.e.
666:where
188:finite
159:) = {
2018:JSTOR
1292:then
204:plane
200:discs
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1579:<
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674:card
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2008:doi
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1543:min
1393:If
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1157:If
813:.
801:of
517:max
483:as
479:of
322:not
313:not
272:.
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53:set
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