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Date: 17-11-2021
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Let be a set and a collection of subsets of . A subset is shattered by if each subset of can be expressed as the intersection of with a subset in . Symbolically, then, is shattered by if for all , there exists some for which .
If is shattered by , one says that shatters .
There are a number of equivalent ways to define shattering. One can easily verify that the above definition is equivalent to saying that shatters if
where denotes the power set of . Yet another way to express this concept is to say that a set of cardinality is shattered by a set if where here,
In the field of machine learning theory, one usually considers the set to be a sample of outcomes drawn according to a distribution with the set representing a collection of "known" concepts or laws. In this context, saying that is shattered by intuitively means that all of the constituent outcomes in can be known by knowing only the laws in .
REFERENCES:
Bhaskar, A. and Sukhar, I. "VC-Dimension." 2008. http://www.cs.cornell.edu/courses/cs683/2008sp/lecture%20notes/683notes_0428.pdf.
Shashua, A. "Lecture 11: PAC II." 2009. http://www.cs.huji.ac.il/~shashua/papers/class11-PAC2.pdf.
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