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In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane ) by thinking of one set of points as being colored blue and the other set of points as being colored red.
Kirchberger's theorem is a theorem in discrete geometry, on linear separability.The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the ...
The Peres–Horodecki criterion is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2×2 and 2×3 dimensional cases the condition is also sufficient.
Linear classification in this non-linear space is then equivalent to non-linear classification in the original space. The most commonly used example of this is the kernel Fisher discriminant . LDA can be generalized to multiple discriminant analysis , where c becomes a categorical variable with N possible states, instead of only two.
Linear separability, a geometric property of a pair of sets of points in Euclidean geometry; Recursively inseparable sets, in computability theory, pairs of sets of natural numbers that cannot be "separated" with a recursive set
Linear separability is testable in time ((/), (), ()), where is the number of data points, and is the dimension of each point. [ 35 ] If the training set is linearly separable, then the perceptron is guaranteed to converge after making finitely many mistakes. [ 36 ]
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.