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HYPERPLANE SEPARATION-THEOREM

Hyperplane separation theorem

On the existence of hyperplanes separating disjoint convex sets

In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar

Hyperplane separation theorem

Hyperplane_separation_theorem

Hahn–Banach theorem

Theorem on extension of bounded linear functionals

Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. The theorem is

Hahn–Banach theorem

Hahn–Banach_theorem

Hyperplane

Subspace of n-space whose dimension is (n-1)

separated by a hyperplane, a result called the hyperplane separation theorem. In machine learning, hyperplanes are a key tool to create support vector machines

Hyperplane

Separation theorem

Index of articles associated with the same name

subset containing one but disjoint from the other. Hyperplane separation theorem - either of two theorems about disjoint convex sets in n-dimensional Euclidean

Separation theorem

Separation_theorem

Supporting hyperplane

Hyperplane in geometry

half-space is the half-space that includes the points within the hyperplane. This theorem states that if S {\displaystyle S} is a convex set in the topological

Supporting hyperplane

Supporting_hyperplane

Farkas' lemma

Solvability theorem for finite systems of linear inequalities

a hyperplane separating the vector from the cone; there are no other possibilities. The closedness condition is necessary, see Separation theorem I in

Farkas' lemma

Farkas'_lemma

Karush–Kuhn–Tucker conditions

Concept in mathematical optimization

m\right\}} , the proof of the Karush–Kuhn–Tucker theorem makes use of the hyperplane separation theorem. The system of equations and inequalities corresponding

Karush–Kuhn–Tucker conditions

Karush–Kuhn–Tucker_conditions

Convex polygon

Polygon that is the boundary of a convex set

corners of the polygon to recover the entire polygon shape. Hyperplane separation theorem: Any two convex polygons with no points in common have a separator

Convex polygon

Convex_polygon

Linear separability

Geometric property of a pair of sets of points in Euclidean geometry

maximize their distance. Clustering (statistics) Hyperplane separation theorem Kirchberger's theorem Perceptron Vapnik–Chervonenkis dimension Boyd, Stephen;

Linear separability

Linear_separability

Kirchberger's theorem

Hyperplane separation theorem, the theorem that disjoint compact convex sets are linearly separable Watson, Donald (1973), "A refinement of theorems of

Kirchberger's theorem

Kirchberger's_theorem

Disjoint sets

Sets with no element in common

pair of the element and the index of the set that contains it. Hyperplane separation theorem for disjoint convex sets Mutually exclusive events Relatively

Disjoint sets

Disjoint_sets

Guillotine cutting

Process of producing small rectangular items of fixed dimensions

total weight can be separated. See also: Geometric separator Hyperplane separation theorem Some recently studied variants of the problem include: Guillotine-cutting

Guillotine cutting

Guillotine_cutting

Dual cone and polar cone

Concepts in convex analysis

the smallest convex cone containing C (a consequence of the hyperplane separation theorem) A cone C in a vector space X is said to be self-dual if X can

Dual cone and polar cone

Dual_cone_and_polar_cone

Gilbert–Johnson–Keerthi distance algorithm

Method of determining minimum distance between two convex sets

NearestSimplex(s) if contains_origin: accept Minkowski Portal Refinement Hyperplane separation theorem Montanari, Mattia; Petrinic, Nik; Barbieri, Ettore (30 June

Gilbert–Johnson–Keerthi distance algorithm

Gilbert–Johnson–Keerthi_distance_algorithm

Hypersurface

Manifold or algebraic variety of dimension n in a space of dimension n+1

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety

Hypersurface

Decision boundary

Hypersurface used by a classification algorithm

the decision boundary and data points. Discriminant function Hyperplane separation theorem Corso, Jason J. (Spring 2013). "Quiz 1 of 14 - Solutions" (PDF)

Decision boundary

Decision_boundary

Meier Eidelheit

Polish mathematician (1910–1943)

versions of the hyperplane separation theorem are also known (in German) as Trennungssatz von Eidelheit (Eidelheit separation theorem). A theorem on the solubility

Meier Eidelheit

Meier_Eidelheit

List of theorems

Hironaka theorem (algebraic geometry) Hodge index theorem (algebraic surfaces) Katz–Lang finiteness theorem (number theory) Lefschetz hyperplane theorem (algebraic

List of theorems

List_of_theorems

Auerbach's lemma

Theorem in functional analysis

hyperplane P n {\displaystyle P_{n}} supporting V {\displaystyle V} at e n {\displaystyle e_{n}} . This is a consequence of the hyperplane separation

Auerbach's lemma

Auerbach's_lemma

Vector-valued Hahn–Banach theorems

Hahn–Banach theorem – Theorem on extension of bounded linear functionals Hyperplane separation theorem – On the existence of hyperplanes separating disjoint

Vector-valued Hahn–Banach theorems

Vector-valued_Hahn–Banach_theorems

Radon's theorem

Theorem in geometry about convex sets

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that: Any set of d + 2 points in Rd can be partitioned into two

Radon's theorem

Radon's_theorem

Convexity in economics

Significant topic in economics

{\displaystyle S,} then there exists a supporting hyperplane containing x . {\displaystyle x.} The hyperplane in the theorem may not be unique, as noticed in the second

Convexity in economics

Convexity_in_economics

Algorithmic problems on convex sets

find a point in K. Strong separation problem (SSEP): given a vector y in Rn, decide whether y in K, and if not, find a hyperplane that separates y from K

Algorithmic problems on convex sets

Algorithmic_problems_on_convex_sets

Oriented matroid

Abstraction of ordered linear algebra

arrangements of hyperplanes). Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma

Oriented matroid

Oriented_matroid

Projective variety

Algebraic variety in a projective space

projective spaces Adequate equivalence relation Hilbert scheme Lefschetz hyperplane theorem Minimal model program Kollár & Moduli, Ch I. Shafarevich, Igor R.

Projective variety

Projective_variety

Homography

Isomorphism of projective spaces in geometry

general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective

Homography

Ellipsoid method

Iterative method for minimizing convex functions

poly(Size(p)) arithmetic operations: A separation oracle for G (that is: either assert that x is in G, or return a hyperplane separating x from G). A first-order

Ellipsoid method

Ellipsoid_method

Banach space

Normed vector space that is complete

called a norming functional for x . {\displaystyle x.} The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space

Banach space

Banach_space

Convex analysis

Mathematics of convex functions and sets

supporting hyperplanes, mixed volumes, and related geometric inequalities. Classical finite-dimensional results such as Carathéodory's theorem (convex hull)

Convex analysis

Convex_analysis

Glossary of functional analysis

the seminorm. Geometrically, it is a generalization of the hyperplane separation theorem. Heine A topological vector space is said to have the Heine–Borel

Glossary of functional analysis

Glossary_of_functional_analysis

Geometric separator

(n − 1)-dimensional hyperplane. Guillotine separation: the problem of separating convex objects in the plane using guillotine cuts. Other Separation theorems. Simultaneous

Geometric separator

Geometric_separator

Perceptron

Algorithm for supervised learning of binary classifiers

positive examples cannot be separated from the negative examples by a hyperplane, then the algorithm would not converge since there is no solution. Hence

Perceptron

Dirac delta function

Generalized function whose value is zero everywhere except at zero

transform because it recovers the value of φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right

Dirac delta function

Dirac_delta_function

Well-posed problem

Property of differential equations describing physical phenomena

analytic functions and a certain transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be

Well-posed problem

Well-posed_problem

Grover's algorithm

Quantum search algorithm

The operator U ω {\displaystyle U_{\omega }} is a reflection at the hyperplane orthogonal to | ω ⟩ {\displaystyle |\omega \rangle } for vectors in the

Grover's algorithm

Grover's_algorithm

Time-translation symmetry

Mathematical transformation in physics

history. Time-translation symmetry is closely connected, via Noether's theorem, to conservation of energy. In mathematics, the set of all time translations

Time-translation symmetry

Time-translation_symmetry

Energy condition

Mathematics of general relativity

(and required) in proofs of various important theorems about black holes, such as the no hair theorem or the laws of black hole thermodynamics. In general

Energy condition

Energy_condition

Hausdorff dimension

Invariant measure of fractal dimension

disjoint. The open set condition is a separation condition that ensures the images ψi(V) do not overlap "too much". Theorem. Suppose the open set condition

Hausdorff dimension

Hausdorff_dimension

Lazy caterer's sequence

Counts pieces of a disk cut by lines

lines; for generalizations to higher dimensions, see arrangement of hyperplanes. The analogue of this sequence in three dimensions is the cake numbers

Lazy caterer's sequence

Lazy_caterer's_sequence

Manifold

Topological space that locally resembles Euclidean space

points can be characterized as those points which land on the boundary hyperplane ( x n = 0 ) {\displaystyle (x_{n}=0)} of R + n {\displaystyle \mathbb

Manifold

Linear discriminant analysis

Method used in statistics, pattern recognition, and other fields

corresponding x → {\displaystyle {\vec {x}}} is located on a certain side of a hyperplane perpendicular to w → {\displaystyle {\vec {w}}} . The location of the

Linear discriminant analysis

Linear_discriminant_analysis

Legendre transformation

Mathematical transformation

of its supporting hyperplanes. This can be seen as consequence of the following two observations. On the one hand, the hyperplane tangent to the epigraph

Legendre transformation

Legendre_transformation

Separoid

Relation on disjoint pairs of sets

with a family of convex sets in some Euclidean space and their separations by hyperplanes. Strausz, Ricardo (1 March 2007). "Homomorphisms of separoids"

Separoid

Glossary of economics

spending. fiscal theory of the price level Fisher equation Fisher separation theorem fixed costs Costs that have to be paid even if a firm is not producing

Glossary of economics

Glossary_of_economics

Wormhole

Hypothetical topological feature of spacetime

{\displaystyle u>0} and u < 0 {\displaystyle u<0} , which are joined by a hyperplane r = 2 m {\displaystyle r=2m} or u = 0 {\displaystyle u=0} in which g {\displaystyle

Wormhole

Four-dimensional space

Geometric space with four dimensions

four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy

Four-dimensional space

Four-dimensional_space

Lonely runner conjecture

Unsolved problem in mathematics

{\displaystyle s<(n-1)/(n+1)} , ignoring rays lying in one of the coordinate hyperplanes. For example, placed in 2-dimensional space, squares any smaller than

Lonely runner conjecture

Lonely_runner_conjecture

120-cell

Four-dimensional analog of the dodecahedron

120-cell beginning with a cell (hull 1). A section is a flat 3-dimensional hyperplane slice through the 3-sphere: a 2-sphere (ordinary sphere). It is dimensionally

120-cell

Multiclass classification

Problem in machine learning and statistical classification

maximizing the margin i.e. maximizing the minimum distance from the separating hyperplane to the nearest example. The basic SVM supports only binary classification

Multiclass classification

Multiclass_classification

Knizhnik–Zamolodchikov equations

Partial differential equations of correlation functions

theory of affine Lie algebra and quantum groups Braid groups Topology of hyperplane complements Knot theory and 3-folds Quantum KZ equations Kohno 2002 Ribault

Knizhnik–Zamolodchikov equations

Knizhnik–Zamolodchikov_equations

Geometry Festival

American annual mathematics conference

University) - The Helly geometry of some fundamental groups of complex hyperplane arrangement complements Wenshuai Jiang (Zhejiang University) - Gromov–Hausdorff

Geometry Festival

Geometry_Festival

Spacetime

Mathematical model combining space and time

{\displaystyle \Delta {d}} between two points can be defined using the Pythagorean theorem: ( Δ d ) 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 {\displaystyle (\Delta {d})^{2}=(\Delta

Spacetime

History of combinatorics

2307/2319793. JSTOR 2319793. Stanley, Richard (2007). "An introduction to hyperplane arrangements". Geometric Combinatorics. IAS/Park City Mathematics Series

History of combinatorics

History_of_combinatorics

Index of physics articles (H)

HT-7 High-temperature engineering test reactor Haag's theorem Haag–Lopuszanski–Sohnius theorem Hadamard–Rybczynski equation Hadron Hadron Elektron Ring

Index of physics articles (H)

Index_of_physics_articles_(H)

Light front quantization

Technique in computational quantum field theory

leave a three-dimensional hyperplane tangent to the light cone invariant. A light front is a three-dimensional hyperplane defined by the condition: with

Light front quantization

Light_front_quantization

Decision tree model

Model of computational complexity

the space is divided into semialgebraic sets (a generalization of a hyperplane). These decision tree models, defined by Rabin and Reingold, are often

Decision tree model

Decision_tree_model

Glossary of artificial intelligence

List of concepts in artificial intelligence

of choosing a set of optimal hyperparameters for a learning algorithm. hyperplane A decision boundary in machine learning classifiers that partitions the

Glossary of artificial intelligence

Glossary_of_artificial_intelligence

Cross-validation (statistics)

Statistical model validation technique

.., xip. If least squares is used to fit a function in the form of a hyperplane ŷ = a + βTx to the data (xi, yi) 1 ≤ i ≤ n, then the fit can be assessed

Cross-validation (statistics)