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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 ...
This means that, given a sample of fixed size , for larger dimensionality it is more probable that a random set of labelled points is linearly separable. Conversely, with fixed dimensionality, for larger sample sizes the number of linearly separable sets of random points will be smaller, or in other words the probability to find a linearly ...
Separable filter, a product of two or more simple filters in image processing; Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals; Separable partial differential equation, a class of equations that can be broken down into differential equations in fewer independent variables
Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset.
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 ]
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
Illustration of a non-convex set. The line segment joining points x and y partially extends outside of the set, illustrated in red, and the intersection of the set with the line occurs in two places, illustrated in black. In geometry, a set of points is convex if it contains every line segment between two points in the set.