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267:. 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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362:, we find that we can still isolate all the subsets from above, as well as the points that were formerly problematic. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below:
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967:, 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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667:{\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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1788:. 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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42:, also known as VC-theory. Shattering and VC-theory are used in the study of
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217:(two-dimensional space) does not shatter every set of four points on the
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1715:{\displaystyle VC_{0}(C)={\underset {n}{\max }}\{n:S_{C}(n)=2^{n}\}.\,}
1609:{\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).
975:. It follows from this that S_C(3) would also be less than 8 (i.e.
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901:, its shattering coefficient(3) would be 8 and S_C(2) would be
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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
1995:, 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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759:{\displaystyle x_{1},x_{2},\dots ,x_{n}\in \Omega }
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1920:A class with finite VC dimension is called a
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1993:Combinatorial Entropy and Uniform Limit Laws
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986:This example illustrates some properties of
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2047:Origin of "Shattered sets" terminology
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690:{\displaystyle \operatorname {card} }
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934:. However, if one of those sets in
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209:We will show that the class of all
1781:{\displaystyle VC(C)=VC_{0}(C)+1.}
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1270:{\displaystyle S_{C}(N)<2^{N}}
403:(visualize connecting the dots).
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452:{\displaystyle s\subset \Omega }
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40:Vapnik–Chervonenkis theory
27:Notion in computational learning
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1394:Vapnik–Chervonenkis class
421:of sets, we use the concept of
390:Each subset of three points in
271:using intersections with class
1902:{\displaystyle S_{C}(n)=2^{n}}
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18:Shattering (machine learning)
1980:10.1016/0012-365X(81)90274-0
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820:with the sets in collection
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479:being any space, often a
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132:{\displaystyle a=c\cap A.}
46:as well as in statistical
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801:{\displaystyle S_{C}(n)}
354:However, if we redefine
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1508:{\displaystyle 2^{0}=1}
1408:cannot be shattered by
1371:{\displaystyle n\geq N}
927:{\displaystyle 2^{2}=4}
886:{\displaystyle 2^{3}=8}
472:{\displaystyle \Omega }
423:shattering coefficients
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2049:, by J. Steele
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2041:External links
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2017:(1): 118â227,
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1973:(3): 313â318,
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700:
684:
658:
655:
652:
649:
646:
643:
635:
631:
627:
624:
621:
616:
612:
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584:
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571:
567:
563:
560:
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552:
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523:
517:
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505:
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489:
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482:
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2014:
2008:
1992:
1970:
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1919:
1914:
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1857:
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1849:
1847:
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1517:VC dimension
1479:
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1400:VC dimension
1387:
1383:
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767:
697:denotes the
676:
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481:sample space
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152:intersection
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32:intersection
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1519:of a class
1443:because as
860:, contains
770:points,.
699:cardinality
401:convex sets
227:unit circle
223:convex sets
219:unit circle
150:when their
2057:Categories
1957:References
1928:. A class
1796:such that
1725:Note that
101:such that
54:Definition
1363:≥
1277:for some
1153:Ω
1150:⊆
1115:⊆
1106:∈
1092:∩
1052:≤
897:shatters
754:Ω
751:∈
735:…
656:∈
644:∩
625:…
588:
580:Ω
577:∈
561:…
532:∀
467:Ω
447:Ω
444:⊂
259:shatters
160:power set
146:shatters
121:∩
2003:(1978),
1991:(1975),
1940:See also
1926:VC class
1909:for all
1352:for all
1139:for any
1078:because
1074:for all
433:of sets
247:The set
81:the set
79:shatters
58:Suppose
2033:2242865
1860:, then
275:and so
213:in the
205:Example
2031:
1792:(i.e.
677:where
199:finite
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215:plane
211:discs
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70:is a
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685:card
585:card
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186:}.
66:and
2019:doi
1975:doi
1932:is
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1228:If
1168:If
824:.
812:of
528:max
494:as
490:of
333:not
324:not
283:.
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