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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 ...
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The left image shows 100 points in the two dimensional real space, labelled according to whether they are inside or outside the circular area. These labelled points are not linearly separable, but lifting them to the three dimensional space with the kernel trick, the points becomes linearly separable. Note that in this case and in many other ...
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 ]
The "trouble" with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space. A first-countable , separable Hausdorff space (in particular, a separable metric space) has at most the continuum cardinality c {\displaystyle {\mathfrak {c}}} .
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 ...