Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, linear separability is a property of two sets of

points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...

. This is most easily visualized in two dimensions (the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

) by thinking of one set of points as being colored blue and the other set of points as being colored red. These two sets are ''linearly separable'' if there exists at least one

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

in the plane with all of the blue points on one side of the line and all the red points on the other side. This idea immediately generalizes to higher-dimensional Euclidean spaces if the line is replaced by a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...

. The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are, arises in several areas. In statistics and

machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

, classifying certain types of data is a problem for which good algorithms exist that are based on this concept.

Mathematical definition

Let

X_

and

X_

be two sets of points in an ''n''-dimensional Euclidean space. Then

X_

and

X_

are ''linearly separable'' if there exist ''n'' + 1 real numbers

w_, w_,..,w_, k

, such that every point

x \in X_

satisfies

\sum^_ w_x_ > k

and every point

x \in X_

satisfies

\sum^_ w_x_ < k

, where

x_

is the

i

-th component of

x

. Equivalently, two sets are linearly separable precisely when their respective convex hulls are

disjoint Disjoint may refer to: *Disjoint sets, sets with no common elements *Mutual exclusivity, the impossibility of a pair of propositions both being true See also *Disjoint union *Disjoint-set data structure {{disambig