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| In [[geometry]], two sets of points in a two-dimensional space are '''linearly separable''' if they can be completely separated by a single line. In general, two point sets are linearly separable in ''n''-dimensional space if they can be separated by a [[hyperplane]].
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| In more mathematical terms: Let <math>X_{0}</math> and <math>X_{1}</math> be two sets of points in an n-dimensional space. Then <math>X_{0}</math> and <math>X_{1}</math> are linearly separable if there exists n+1 real numbers <math>w_{1}, w_{2},..,w_{n}, k</math>, such that every point <math>x \in X_{0}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i}\ge k</math> and every point <math>x \in X_{1}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i} < k</math>, where <math>x_{i}</math> is the <math>i</math>-th component of <math>x</math>.
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| Equivalently, two sets are linearly separable precisely when their respective [[convex hull]]s are [[disjoint sets|disjoint]] (colloquially, do not overlap).
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| == Example ==
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| Three points in two classes ('+' and '-') are always linearly separable in two dimensions. This is illustrated by the three examples in the following figure:
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| {| align="center" border="0" cellpadding="4" cellspacing="0"
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| | align="center" | [[File:VC1.svg]]
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| | align="center" | [[File:VC2.svg]]
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| | align="center" | [[File:VC3.svg]]
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| |}
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| However, not all sets of four points are linearly separable in two dimensions. The following example would need '''two''' straight lines and thus is not linearly separable:
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| {| align="center" border="0" cellpadding="4" cellspacing="0"
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| | [[File:VC4.svg]]
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| |}
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| == Linear separability of hypercubes in n dimensions ==
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| {| class="wikitable"
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| |+<small>Number of linearly separable Boolean hypercubes in each dimension</small><ref>
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| {{cite paper
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| | last=Gruzling
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| | first=Nicolle
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| | title=Linear separability of the vertices of an n-dimensional hypercube. M.Sc Thesis
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| | publisher= University of Northern British Columbia
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| | year=2006
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| }}</ref> {{OEIS|id=A000609}}
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| !Dimension
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| !Linearly separable Boolean hypercubes
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| |-
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| | 2 || 14
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| |-
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| | 3 || 104
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| |-
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| | 4 || 1882
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| |-
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| | 5 || 94572
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| |-
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| | 6 || 15028134
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| |-
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| | 7 || 8378070864
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| |-
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| | 8 || 17561539552946
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| |-
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| | 9 || 144130531453121108
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| |}
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| == Usage ==
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| Linear separability allows simple [[Classification in machine learning]].
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| == See also ==
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| * [[Vapnik–Chervonenkis dimension]]
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| == References ==
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| {{reflist}}
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| [[Category:Multi-dimensional geometry]]
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| [[Category:Convex analysis]]
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| {{Geometry-stub}}
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