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Hyperplane Separation Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separating Axis Theorem2008
Separation may refer to: Films * ''Separation'' (1967 film), a British feature film written by and starring Jane Arden and directed by Jack Bond * '' La Séparation'', 1994 French film * ''A Separation'', 2011 Iranian film * ''Separation'' (2013 film), an American thriller film starring Sarah Manninen * ''Separation'' (2021 film), an American horror film Literature * ''The Separation'' (Bury Novel), an 1830 novel by Lady Charlotte Bury * ''The Separation'' (Priest novel), a 2002 novel by Christopher Priest * ''The Separation'' (Applegate novel), a 1999 novel in the ''Animorphs'' series by K.A. Applegate * ''Separation'', a 1976 Canadian political novel by Richard Rohmer Music * ''Separation'', a 2011 album by Balance and Composure * ''Separation'' (EP), a 2006 EP by Halou * ''Separations'' (album), a 1992 album by British alternative rock band Pulp * Separation mastering, in music recording Law and politics * Marital separation, when a married couple ceases living tog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support-vector Machine
In machine learning, support vector machines (SVMs, also support vector networks) are supervised learning, supervised Maximum-margin hyperplane, max-margin models with associated learning algorithms that analyze data for Statistical classification, classification and regression analysis. Developed at AT&T Bell Laboratories, SVMs are one of the most studied models, being based on statistical learning frameworks of VC theory proposed by Vladimir Vapnik, Vapnik (1982, 1995) and Alexey Chervonenkis, Chervonenkis (1974). In addition to performing linear classifier, linear classification, SVMs can efficiently perform non-linear classification using the Kernel method#Mathematics: the kernel trick, ''kernel trick'', representing the data only through a set of pairwise similarity comparisons between the original data points using a kernel function, which transforms them into coordinates in a higher-dimensional feature space. Thus, SVMs use the kernel trick to implicitly map their inputs int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the Closure (topology), closure of not belonging to the Interior (topology), interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a Manifold#Manifold with boundary, different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Felix Hausdorff, Hausdorff's border, which is defined as the intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Interior
In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set S (denoted \operatorname(S)) is defined as its interior within the affine hull of S. In other words, \operatorname(S) := \, where \operatorname(S) is the affine hull of S, and B_\epsilon(x) is a ball of radius \epsilon centered on x. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior. A set is relatively open iff it is equal to its relative interior. Note that when \operatorname(S) is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed. For any convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Set
In mathematics, an affine space is a geometry, geometric structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance (mathematics), distance and measure of angles, keeping only the properties related to parallel (geometry), parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called ''point (geometry), points'', which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane (geometry), plane can be drawn; and, in general, through points in general position, a -dimensional flat (geometry), flat or affine subspace can be drawn. Affine space is charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Sum
In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowski decomposition'', or ''geometric difference'') is the corresponding inverse, where (A - B) produces a set that could be summed with ''B'' to recover ''A''. This is defined as the complement of the Minkowski sum of the complement of ''A'' with the reflection of ''B'' about the origin. \begin -B &= \\\ A - B &= (A^\complement + (-B))^\complement \end This definition allows a symmetrical relationship between the Minkowski sum and difference. Note that alternately taking the sum and difference with ''B'' is not necessarily equivalent. The sum can fill gaps which the difference may not re-open, and the difference can erase small islands which the sum cannot recreate from nothing. \begin (A - B) + B &\subseteq A\\ (A + B) - B &\supseteq A\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Interior
In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set S (denoted \operatorname(S)) is defined as its interior within the affine hull of S. In other words, \operatorname(S) := \, where \operatorname(S) is the affine hull of S, and B_\epsilon(x) is a ball of radius \epsilon centered on x. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior. A set is relatively open iff it is equal to its relative interior. Note that when \operatorname(S) is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed. For any convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separating Hyperplanes Theorem
Separation may refer to: Films * ''Separation'' (1967 film), a British feature film written by and starring Jane Arden and directed by Jack Bond * '' La Séparation'', 1994 French film * ''A Separation'', 2011 Iranian film * ''Separation'' (2013 film), an American thriller film starring Sarah Manninen * ''Separation'' (2021 film), an American horror film Literature * ''The Separation'' (Bury Novel), an 1830 novel by Lady Charlotte Bury * ''The Separation'' (Priest novel), a 2002 novel by Christopher Priest * ''The Separation'' (Applegate novel), a 1999 novel in the ''Animorphs'' series by K.A. Applegate * ''Separation'', a 1976 Canadian political novel by Richard Rohmer Music * ''Separation'', a 2011 album by Balance and Composure * ''Separation'' (EP), a 2006 EP by Halou * ''Separations'' (album), a 1992 album by British alternative rock band Pulp * Separation mastering, in music recording Law and politics * Marital separation, when a married couple ceases living tog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn–Banach Theorem
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. History The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C , b/math> of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Separation Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
