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Hyperplane in geometry
In geometry, a supporting hyperplane of a set S {\displaystyle S} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a hyperplane that has both
Supporting_hyperplane
Subspace of n-space whose dimension is (n-1)
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like
Hyperplane
On the existence of hyperplanes separating disjoint convex sets
is the supporting hyperplane theorem. In the context of support-vector machines, the optimally separating hyperplane or maximum-margin hyperplane is a hyperplane
Hyperplane_separation_theorem
Topics referred to by the same term
measurable space Supporting hyperplane, sometimes referred to as support Support of a module, a set of prime ideals in commutative algebra Support, the natural
Support
Significant topic in economics
points in Q. Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set S {\displaystyle
Convexity_in_economics
Distance from origin of tangent hyperplanes
^{n}} describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on R n {\displaystyle
Support_function
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
Hahn–Banach_theorem
Set of methods for supervised statistical learning
perceptron of optimal stability. More formally, a support vector machine constructs a hyperplane or set of hyperplanes in a high or infinite-dimensional space,
Support_vector_machine
analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set. Let X be a locally convex topological
Supporting_functional
In geometry, set whose intersection with every line is a single line segment
convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set C and point
Convex_set
Convex hull of a finite set of points in a Euclidean space
corresponds with a supporting hyperplane of the polytope, a hyperplane bounding a half-space that contains the polytope. If a supporting hyperplane also intersects
Convex_polytope
interior. The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane. If two bounded connected
Supporting_line
Concept in mathematical optimization
\mathbf {\alpha } )} . Since the idea of this approach is to find a supporting hyperplane on the feasible set Γ = { x ∈ X : g i ( x ) ≤ 0 , i = 1 , … , m
Karush–Kuhn–Tucker_conditions
Mathematics of convex functions and sets
minimum. Convex sets can often be separated by hyperplanes, and convex functions can be studied through supporting affine functions. Convex analysis is a common
Convex_analysis
Steinitz theorem (graph theory) Stewart's theorem (plane geometry) Supporting hyperplane theorem (convex geometry) Sylvester–Gallai theorem (plane geometry)
List_of_theorems
Mathematical transformation
terms of its supporting hyperplanes. This can be seen as consequence of the following two observations. On the one hand, the hyperplane tangent to the
Legendre_transformation
Economic Model
price hyperplane ⟨ p , q ⟩ = ⟨ p , ∑ j y j ⟩ {\displaystyle \langle p,q\rangle =\langle p,\sum _{j}y^{j}\rangle } . Since it's a supporting hyperplane of
Arrow–Debreu_model
Concepts in convex analysis
is a normal at the origin of a hyperplane that supports C. y and C lie on the same side of that supporting hyperplane. C* is closed and convex. C 1 ⊆
Dual_cone_and_polar_cone
not differentiable Supporting hyperplane - a hyperplane meeting certain conditions Supporting hyperplane theorem - that defines a supporting hyperplane
List_of_convexity_topics
Russian mathematician (born 1966)
exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes.[P89] As such, his construction provided further obstruction to
Grigori_Perelman
Branch of applied mathematics
particularly by clarifying the role of prices as normal vectors to a supporting hyperplane of a convex set representing production or consumption possibilities
Mathematical_economics
Generalization of the Legendre transformation
encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. For more examples, see § Table of selected convex conjugates. The
Convex_conjugate
Generalization of the tangent space to a manifold to the case of certain spaces
{\displaystyle V} containing K {\displaystyle K} and bounded by the supporting hyperplanes of K {\displaystyle K} at x {\displaystyle x} . The boundary T K
Tangent_cone
Hypersurface used by a classification algorithm
output label of a classifier is ambiguous. If the decision surface is a hyperplane, then the classification problem is linear, and the classes are linearly
Decision_boundary
Machine learning strategy
n-dimensional distance from that datum to the separating hyperplane. Minimum Marginal Hyperplane methods assume that the data with the smallest W are those
Active learning (machine learning)
Active_learning_(machine_learning)
Mathematical set closed under positive linear combinations
given each linear form associated with the halfspaces also define a support hyperplane of a facet. Each face of a polyhedral cone is spanned by some subset
Convex_cone
Geometric property of a pair of sets of points in Euclidean geometry
is replaced by a hyperplane. The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are, arises
Linear_separability
to the high- or infinite-dimensional space. In case such a separating hyperplane does not exist, we introduce so-called slack variables ξ i {\displaystyle
Least-squares support vector machine
Least-squares_support_vector_machine
Mathematical problem
convex body C in Rn and a hyperplane H, the width of C parallel to H, w(C,H), is the distance between the two supporting hyperplanes of C that are parallel
Tarski's_plank_problem
Distance from a data point to a decision boundary
should choose the hyperplane such that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known
Margin_(machine_learning)
Geometric transformation combining reflection and translation
consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation
Glide_reflection
Minkowsi sum of line segments
as a projection of a hypercube. Zonotopes are intimately connected to hyperplane arrangements and matroid theory. The Minkowski sum of a finite set of
Zonotope
Multidimensional search tree for points in k dimensional space
generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the
K-d_tree
Loss function in machine learning
( w , b ) {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and x {\displaystyle \mathbf {x} } is the input variable(s). When t and
Hinge_loss
Concept in algebraic geometry
a hyperplane in P n {\displaystyle \mathbb {P} ^{n}} (because the zero set of a section of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is a hyperplane).
Ample_line_bundle
Mathematical object
intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection
3-sphere
Mathematical algorithm
then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search
Coordinate_descent
and Vladimir Vapnik, and framed geometrically as a method for finding hyperplanes that can separate multidimensional data into two categories. This traditional
Regularization perspectives on support vector machines
Regularization_perspectives_on_support_vector_machines
Conic solid with a polygonal base
− 1)-polytope in a (n − 1)-dimensional hyperplane. A point called the apex is located outside the hyperplane and gets connected to all the vertices of
Pyramid_(geometry)
Set of statistical processes for estimating the relationships among variables
the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical
Regression_analysis
Concept in projective geometry
pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double
Duality_(projective_geometry)
Abstraction of ordered linear algebra
properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary (i.e., non-oriented)
Oriented_matroid
Approximations used in machine learning
or infinite-dimensional feature space and find the optimal splitting hyperplane. In the kernel method the data is represented in a kernel matrix (or,
Low-rank matrix approximations
Low-rank_matrix_approximations
Algorithmic technique using hashing
hyperplane (defined by a normal unit vector r) at the outset and use the hyperplane to hash input vectors. Given an input vector v and a hyperplane defined
Locality-sensitive_hashing
Property of a mathematical space
that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety. An algebraic set being
Dimension
Statistical method
example, the hyperplane is just a 2-dimensional plane defined by the two factor vectors. The projection of the data vectors onto the hyperplane is given by
Factor_analysis
Integral transform in mathematics
{\displaystyle Rf} on the space Σ n {\displaystyle \Sigma _{n}} of all hyperplanes in R n {\displaystyle \mathbb {R} ^{n}} . It is defined by: R f ( ξ )
Radon_transform
Hungarian and American mathematician and physicist (1903–1957)
vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been primary
John_von_Neumann
machine learning Support Vector Machines (SVM) – algorithm for finding a hyperplane to separate classes ADaMSoft ADMB Chronux DAP Epi Info Fityk GNU Octave
List_of_data_science_software
Probability distribution
on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. The probability density function (PDF) is given by
Folded_normal_distribution
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
Statistical classification in machine learning
for binary classification. Support vector machine—an algorithm that maximizes the margin between the decision hyperplane and the examples in the training
Linear_classifier
Australian and American mathematician (born 1975)
singular integral operators with the multiplier allowed to degenerate on a hyperplane, identifying conditions which ensure operator continuity relative to Lp
Terence_Tao
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
Mathematical condition
\alpha _{1}} denote the restrictions of α {\displaystyle \alpha } to the hyperplanes t = 0 , t = 1 {\displaystyle t=0,t=1} and they are zero since d t {\displaystyle
Poincaré_lemma
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
Gilbert–Johnson–Keerthi distance algorithm
Gilbert–Johnson–Keerthi_distance_algorithm
Operations research that evaluates multiple conflicting criteria in decision making
nondominated set are located either on vertical or horizontal planes (hyperplanes) in the criterion space. Ideal point: (in criterion space) represents
Multiple-criteria decision analysis
Multiple-criteria_decision_analysis
Machine learning kernel function
learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the
Polynomial_kernel
Regression analysis for modeling ordinal data
a variant of the perceptron algorithm that found multiple parallel hyperplanes separating the various ranks; its output is a weight vector w and a sorted
Ordinal_regression
Class of algorithms for pattern analysis
class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to
Kernel_method
Area of mathematical analysis
packets overlap in physical space. Frequencies concentrated on a flat hyperplane do not disperse like frequencies on a curved hypersurface. For curved
Harmonic_analysis
Integral transform
of a function is defined by integrating over lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from
X-ray_transform
Euclidean Wightman distributions
groups of points lie on two sides of the x 0 = 0 {\displaystyle x^{0}=0} hyperplane, while the vector b {\displaystyle b} is parallel to it: x 1 0 , … , x
Schwinger_function
Type of mathematical plane curve
Hedgehogs can also be defined from support functions of hyperplanes in higher dimensions. Formally, a planar support function can be defined as a continuously
Hedgehog_(geometry)
Smooth approximation of one-hot arg max
to the linear constraint that all output sum to 1 meaning it lies on a hyperplane. Along the main diagonal ( x , x , … , x ) , {\displaystyle (x,\,x,\,\dots
Softmax_function
Philosophical argument based on the theory of relativity
relativity the present is a local concept that cannot be extended to global hyperplanes. Furthermore, N. David Mermin states: That no inherent meaning can be
Rietdijk–Putnam_argument
Machine learning problem
distorted probability distribution or the "signed distance to the hyperplane" in a support vector machine). Deviations from the identity function indicate
Probabilistic_classification
Complete, full information, perfectly competitive markets are Pareto efficient
These two convex, non-intersecting sets allow us to apply the separating hyperplane theorem. This theorem states that there exists a price vector p ≠ 0 {\displaystyle
Fundamental theorems of welfare economics
Fundamental_theorems_of_welfare_economics
Broadest definition of sizes in integer-dimensional spaces
{\displaystyle n\geq 2} , has a zero Lebesgue measure. In general, every proper hyperplane has a zero Lebesgue measure in its ambient space. The volume of an n-ball
Lebesgue_measure
the associated distribution of hyperplanes at the point u ∈ UTxM is the inverse image under π* of the tangent hyperplane to the unit sphere in TxM at u
Unit_tangent_bundle
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
Surface in 3D space defined by an implicit function of three variables
function to find the distance to the surface. Open-source or free software supporting algebraic implicit surface modelling: K3DSurf — A program to visualize
Implicit_surface
Type of geometry
affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model for doing
Projective_geometry
Mathematical sequence
\geq 2} (this is because of the Hurewicz homomorphism and the Lefschetz hyperplane theorem). In this case the local systems R q f ∗ ( Q _ X ) {\displaystyle
Leray_spectral_sequence
Euclidean space without distance and angles
– 1 in an affine space or a vector space of dimension n is an affine hyperplane. The following characterization may be easier to understand than the usual
Affine_space
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
Real function with finite total variation
this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed
Bounded_variation
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
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
Tree-based ensemble machine learning methods
in 1995. Ho established that forests of trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as
Random_forest
training-set and test-set) Support Vector Machine (SVM): a set of methods which divide multidimensional data by finding a dividing hyperplane with the maximum margin
List_of_algorithms
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)
Theorem in geometry
∈ l {\textstyle t\in l} let H t {\textstyle H_{t}} denote the affine hyperplane orthogonal to l {\textstyle l} that passes through t {\textstyle t} .
Brunn–Minkowski_theorem
Type of vector space in math
closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation
Hilbert_space
Problem in machine learning and statistical classification
concerned. Support vector machines are based upon the idea of maximizing the margin i.e. maximizing the minimum distance from the separating hyperplane to the
Multiclass_classification
Dolbeault cohomology of compact Kähler manifolds, such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations
Kähler_identities
Subfield of mathematical optimization
Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.[citation needed] The convex programs easiest
Convex_optimization
Concept in mathematics
preceding example, let U = A1 − {1}. Since U is the complement of the hyperplane t = 1, U is affine. The restriction f : U → X {\displaystyle f:U\to X}
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
In mathematics, vector space of linear forms
parallel hyperplanes in V {\displaystyle V} , and the action of a linear functional on a vector can be visualized in terms of these hyperplanes. If V {\displaystyle
Dual_space
Machine learning algorithm
Euclidean, though others may be used) of a sample from the separating hyperplane is the margin of that sample. The notion of margins is important in several
Margin_classifier
Theorem in functional analysis
exists a 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
Colombian mathematician
de los Andes, and "Todos Cuentan," a community-based project aimed at supporting mathematicians from underrepresented backgrounds. He has published over
Federico_Ardila
Soviet hypersonic aircraft
has media related to Tupolev Tu-2000. SPACE TRANSPORT: Tupolev Tu-2000 Hyperplane – Russia "Tu-2000". Encyclopedia Astronautica. Mark Wade. Archived from
Tupolev_Tu-2000
Geometric object with flat sides
Regular polytopes, p. 127 The part of the polytope that lies in one of the hyperplanes is called a cell Beck, Matthias; Robins, Sinai (2007), Computing the
Polytope
Database of data representing objects in geometric space
include: Binary space partitioning (BSP-Tree): Subdividing space by hyperplanes. Bounding volume hierarchy (BVH) Geohash Grid (spatial index) HHCode
Spatial_database
Invariant of algebraic varieties and of more general schemes
algebraic cycles on the product of X with affine space which meet a set of hyperplanes (viewed as the faces of a simplex) in the expected dimension. In the
Motivic_cohomology
Extending the elements of a polytope to form a new figure
stellation diagram of an n-polytope exists in an (n − 1)-dimensional hyperplane of a given facet. For example, in 4-space, the great grand stellated 120-cell
Stellation
approach. If X is a smooth projective variety of dimension m and H is a hyperplane section, then a vector bundle (or a torsion-free sheaf) W is called stable
Stable_vector_bundle
Concept in machine learning
whereas points within the margin boundaries or on the wrong side of the hyperplane are penalized in a linear fashion compared to their distance from the
Loss functions for classification
Loss_functions_for_classification
dimensions, one has to consider ( n − 1 ) {\displaystyle (n-1)} -dimensional hyperplanes perpendicular to a given direction n ^ {\displaystyle {\hat {n}}} in
Mean_width
SUPPORTING HYPERPLANE
SUPPORTING HYPERPLANE
Male
Greek
(Ὀφιοῦχος) Greek name OPHIUCHUS means "serpent bearer." This is the name of one of the constellations listed by Ptolemy, depicted as a man supporting a serpent. The man depicted in the constellation is thought by some to actually be the demigod Asklepios.
Girl/Female
Tamil
Adrita | அதà¯à®°à®¿à®¤à®¾
Independent, Supportive, One who is loved by everyone
Adrita | அதà¯à®°à®¿à®¤à®¾
Male
Greek
(ΟφιοÏχος) Greek name OPHIOUCHOS means "serpent bearer." This is the name of a constellation depicted as a man supporting a serpent. The man is thought by some to be the demigod Asklepios, who learned the secret of life and death from a serpent and was killed for this by Zeus to prevent him from sharing his knowledge with mankind.
Surname or Lastname
English
English : from an agent derivative of Middle English pleyen ‘to play’, hence an occupational name for an actor or musician or a nickname for a successful competitor in contests of athletic or sporting prowess.
Boy/Male
Tamil
Supporting
Boy/Male
Indian, Punjabi, Sikh
Supporting People
Boy/Male
Arabic, Australian
Helping; Supporting
Girl/Female
Hindu, Indian
Supportive; Modification of the Name Saranya
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil
Supporting
Surname or Lastname
English
English : metonymic occupational name for a baker, from the Middle English term cocket-bread, denoting a high-quality leavened bread, second only to the wastell or finest bread. It has been suggested that this bread may have derived its name from Anglo-French cockette ‘seal’, having supposedly been marked with the seal of the King’s Custom House, though there is no supporting evidence for this.
Girl/Female
Indian
Independent, Supportive, One who is loved by everyone
Boy/Male
Hindu, Indian, Sanskrit, Traditional
Supporting; Nourishing; Another Name for Vishnu
Boy/Male
Hindu, Indian, Marathi
Delighting; Gratifying; Sporting
Boy/Male
Indian, Punjabi, Sikh
Supporting Love
Girl/Female
Tamil
Supporting
Boy/Male
Hindu
Supporting
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Sporting; An Angel
Boy/Male
Hindu, Indian
Playing; Sporting
Boy/Male
Tamil
Ullasin | உலà¯à®²à®¾à®¸à¯€à®¨
Playing, Sporting
SUPPORTING HYPERPLANE
SUPPORTING HYPERPLANE
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
The One with the Good
Boy/Male
Muslim
Brave, Champion, Hero
Boy/Male
Greek
Lordly.
Boy/Male
Hindu, Indian, Tamil
Power; Goddess Durga
Boy/Male
African, Hebrew, Hindu, Indian, Marathi
Garden of the Lord; Notable; Important Person
Male
Irish
Irish name derived from the Gaelic word biorach, BEARACH means "sharp."
Boy/Male
Muslim
Boy/Male
Muslim
Judge
Girl/Female
Arabic
Short for Yasmin or Yasamin
Girl/Female
Indian
Famous, Good, Pious
SUPPORTING HYPERPLANE
SUPPORTING HYPERPLANE
SUPPORTING HYPERPLANE
SUPPORTING HYPERPLANE
SUPPORTING HYPERPLANE
a.
Producing, having, or supporting nails or claws.
p. pr. & vb. n.
of Purport
a.
Flower bearing; supporting the flower.
p. pr. & vb. n.
of Suppurate
n.
The act of supporting or defending.
p. pr. & vb. n.
of Sport
a.
Of pertaining to, or engaging in, sport or sporrts; exhibiting the character or conduct of one who, or that which, sports.
a.
Supporting; sustaining; as, a sustentacular tissue.
n.
The act of supporting orphans.
n.
A stem, or footstalk, supporting the fruit.
n.
Manner of supporting or continuing life or vegetation.
a.
Helping; aiding; supporting.
n.
A pole for supporting a scaffold.
n.
A bracket supporting a cornice; a console.
p. pr. & vb. n.
of Suppose
n.
The supporting frame of a run of millstones.
p. pr. & vb. n.
of Support
a.
Strengthening; supporting; corroborating.
n.
A framework for supporting a bed.
n.
a bandage or bag for supporting the scrotum.