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Topic: Hyperplane

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	Piecewise Linear Classification with Hyperplanes
		The key concept is to cut as many Tomek links as possible with each hyperplane, given the constraint that they must be able to classify points with an error less that a predefined maximum acceptable classification error.
		If the hyperplane is incapable of locally classifying points with an error rate less than maximum acceptable error threshold, it becomes part of the set of hyperplanes, and a new hyperplane is created on a new Tomek link.
		As in the case with a hyperplane cutting a single Tomek link, the hyperplanes cutting multiple Tomek links are locally trained to classify points on either side with a minimal error rate.
	www.cim.mcgill.ca /~mtoews/plc/algorithm.html (1648 words)

	Homogeneous Transformation Matrices
		The "hyperplane" is the class of all such representations, but for convenience we may identify a particular representation as the hyperplane.
		Hyperplane matrices are represented by lower case letters, or by lower case superscripts.
		The normal of the hyperplane at infinity, w, is undefined.
	www.silcom.com /~barnowl/HTransf.htm (4174 words)

	Deformations of Coxeter Hyperplane Arrangements - Postnikov, Stanley (ResearchIndex)
		Abstract: We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements.
		55 Combinatorics and topology of complements of hyperplanes (context) - Orlik, Solomon - 1980
		1 a family of hyperplane arrangements related to ane Weyl grou..
	citeseer.ist.psu.edu /postnikov00deformations.html (792 words)

	Max Wakefield's Home Page
		Hyperplane Arrangements which is in Vancouver, B.C. from August 20th to August 24th.
		Arrangements of Hyperplanes - Algebra, Combinatorics, Geometry, and Topology was in Ascona, Switzerland from May 15th to May 20th.
		The characteristic polynomial of a multiarrangement of hyperplanes
	www.math.sci.hokudai.ac.jp /~wakefield (815 words)

	The Miracle Octad Generator
		It so happens that the octad stabilizer, which may be viewed as acting separately on the stabilized octad and on the set of the remaining 16 points of the 24, is isomorphic to the automorphism group of the affine 4-space over the 2-element field.
		The group of the geometry of the affine n-space is the subgroup of the group of the projective n-space leaving the "hyperplane at infinity" invariant (as a set, not pointwise), as in Veblen's discussion above.
		One might argue that a stabilized octad in the Mathieu geometry plays a role in some way analogous to that of a "hyperplane at infinity."
	finitegeometry.org /sc/24/MOG.html (827 words)

	Hyperplane Arrangments (Site not responding. Last check: )
		Lecture notes on hyperplane arrangements (114 pages) based on a lecture series at the Park City Mathematics Institute, July 12-19, 2004:
		These notes provide an introduction to hyperplane arrangements, focusing on connections with combinatorics, at the beginning graduate student level.
		After going through these notes a student should be ready to study the deeper algebraic and topological aspects of the theory of hyperplane arrangements.
	www-math.mit.edu /~rstan/arr.html (115 words)

	PlanetMath: hyperplane arrangement
		By selecting a point in each cell and taking the convex hull of the result, we obtain a polytope combinatorially equivalent to the zonotope dual to the arrangement.
		Cross-references: embedding, zonotope, polytope, convex hull, point, cells, contractible, finite, complement, fundamental group, topological vector space, lines, projective space, subspaces, affine subspaces, pass through, hyperplane, field, vector space
		This is version 3 of hyperplane arrangement, born on 2006-03-23, modified 2006-03-23.
	planetmath.org /encyclopedia/HyperplaneArrangement.html (210 words)

	NationMaster - Encyclopedia: Hyperplane (Site not responding. Last check: )
		In a one-dimensional space (such as a line), a hyperplane is a point; it divides a line into two rays.
		In three-dimensional space, a hyperplane is an ordinary plane; it divides the space into two half-spaces.
		In the general case, a hyperplane is an affine subspace of codimension 1.
	www.nationmaster.com /encyclopedia/Hyperplane (643 words)

	Convex Sets (Site not responding. Last check: )
		A hyperplane is the set points of the vector space that map into the same real value; i.e., x such that f(x)=b.
		The hyperplane has associated with it two open half spaces; i.e., the set of points such that f(x)points such that f(x)>b.
		There are also to closed half spaces associated with a hyperplane; i.e., the set of points such that f(x)≤b and the set of points such that f(x)≥b.
	www.sjsu.edu /faculty/watkins/convex.htm (316 words)

	[No title]
		The decision boundary (a hyperplane in n-dimension space) is drawn in such a way that the distance from the closest points of both classes to the hyperplane is maximized.
		The margin of the hyperplane is the sum of the distances between the hyperplane and the nearest points on each side of the hyperplane.
		In particular, the hyperplane with the largest margin, which is defined as the sum of the distances from the hyperplane 'H' to the nearest positive and negative examples is desired.
	www.cs.wisc.edu /~apirak/cs/cs838/reviews_score_9.html (6152 words)

	New Page 2 (Site not responding. Last check: )
		One of the key distinctions between hyperplane and subspace arrangements is the restriction on how hyperplanes intersect.
		Whenever hyperplanes intersect they always intersect in a subspace one dimension less.
		So for an arbitrary intersection of hyperplanes, when another hyperplane is intersected the resulting subspace is either the same or is one dimension less.
	mcs.edgewood.edu /math/jewell/arrangements/realization.htm (973 words)

	Maximising the hyperplane count (with and without division by the square root of the communality)
		That is why, as in Cattell and Muerle (1966), a trial-and-error maximisation of the hyperplane count will be performed with Trasid, however with the difference that in the cases, in which the iterative procedure cannot maximise the hyperplane count (i.e.
		Maximising of P supports the maximisation of the hyperplane count, because the lower loadings are pushed in the direction of the hyperplanes (e.g.
		The maximisation of P thus increases the propability for maximising the combined hyperplane count by means of the trial-and-error-algorithm.
	www.dgps.de /fachgruppen/methoden/mpr-online/issue3/art10/node4.html (2344 words)

	Gist SVM Server
		In theory, a simple way to build a binary classifier is to construct a hyperplane (i.e., a plane in a space with more than three dimensions) separating class members (positive examples) from non-members (negative examples) in this space.
		For some data sets, the SVM may not be able to find a separating hyperplane in feature space, either because the kernel function is inappropriate for the training data or because the data contains mislabeled examples.
		When the magnitude of the noise in the negative examples outweighs the total number of positive examples, the optimal hyperplane located by the SVM will be uninformative, classifying all members of the training set as negative examples.
	svm.sdsc.edu /svm-overview.html (1040 words)

	Support vector machines (Site not responding. Last check: )
		SVMs find the maximum margin hyperplane, the hyperplane that maximixes the minimum distance from the hyperplane to the closest training point (see Figure 2).
		Furthermore, the algorithm that finds a separating hyperplane in the feature space can be stated entirely in terms of vectors in the input space and dot products in the feature space.
		The maximum margin allows the SVM to select among multiple candidate hyperplanes; however, for many data sets, the SVM may not be able to find any separating hyperplane at all, either because the kernel function is inappropriate for the training data or because the data contains mislabelled examples.
	www.cse.ucsc.edu /research/compbio/genex/genexTR2html/node3.html (1205 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: )
		The most familiar kind of hyperplane is an affine hyperplane; that is the kind described here.
		In a one-dimensional space (a straight line), a hyperplane is a point; it divides a line into two rays.
		In the general case, an affine hyperplane is an affine subspace of codimension 1 in an affine space.
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=hyperplane (292 words)

	Reference.com/Encyclopedia/Hyperplane at infinity
		A pair of non-parallel affine hyperplanes intersect at an affine subspace of dimension n − 2, but a parallel pair of affine hyperplanes intersect at a projective subspace of the ideal hyperplane (the intersection lies on the ideal hyperplane).
		Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity.
		In a projective space, any hyperplane may be chosen to be the hyperplane at infinity.
	www.reference.com /browse/wiki/Hyperplane_at_infinity (406 words)

	Convex Structures
		The Separating Hyperplane Theorem is merely a version of the Hahn-Banach Theorem for functional analysis - and really states nothing much more complicated than that two convex sets can be separated by a line ("hyperplane") with one of the convex sets lying on one side of it and the other set sitting on the other.
		Generally, a hyperplane is a linear subspace of one dimension less than the space in which it sits (thus diagramatically, a "hyperplane" is a line drawn in a two-dimensional space or a plane drawn in three-dimensional space).
		Supporting Hyperplane: A hyperplane H is said to be a "supporting hyperplane" to a convex set X if X is contained within one of the halfspaces of H and the boundary of X has points in comon with H, i.e.
	cepa.newschool.edu /het/essays/math/convex.htm (1799 words)

	Binary Space Partition Trees in 3d worlds
		The end goal of a BSP tree if for the hyperplanes of the leaf nodes to be trivially "behind" or "infront" of the parent hyperplane.
		The recursive BSP tree is very simple to understand because it simply performs a partition based on the current hyperplane and then recurses the the front and back leaf nodes.
		This is easy because the hyperplane of this node split the subspace into a empty and non-empty region.
	web.cs.wpi.edu /~matt/courses/cs563/talks/bsp/document.html (2226 words)

	Hyperplane_d<Kernel> (Site not responding. Last check: )
		with respect to the hyperplane (on the hyperplane, on the negative side, or on the positive side).
		In other words, two hyperplanes are strongly equal if their coefficient vectors are positive multiples of each other and they are (weakly) equal if their coefficient vectors are multiples of each other.
		Hyperplanes are implemented by arrays of integers as an item type.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Kernel_d_ref/Class_Hyperplane_d.html (395 words)

	Hyperplane (Site not responding. Last check: )
		In geometry, a hyperplane is a generalisation of a normal two-dimensional plane in three-dimensional space to its (n − 1)-dimensional analogue in n-dimensional space, where n is an arbitrary number.
		Of course, the number of degrees of freedom can be further restricted to produce a hyperplane of a lower number of dimensions (except in the base case where n = 1), but when discussing n-dimensional space the unmodified term "hyperplane" usually denotes an (n − 1)-dimensional hyperplane.
		A zero-dimensional hyperplane is a point; a one-dimensional hyperplane is a (straight) line; and a two-dimensional hyperplane is a plane.
	www.fact-index.com /h/hy/hyperplane.html (192 words)

	Approximate Matrix Inverses and the Condition Number
		Now assume, for purpose of contradiction, that there is a point d in C on the hyperplane or on the same side as x.
		Since the closure of C is convex, the line segment from d to c lies entirely in the closure of C. The accompanying illustration shows the relationship in a two-dimensional plane containing d, c and x.
		The angle dcx is acute, so some point on the line segment sufficiently close to c must be closer to x than c, which is a contradiction, because c was the closest point in the closure of C to x.
	www.efgh.com /math/invcond.htm (1135 words)

	India enters the cruise missile race; hyperplane Avatar reaches planning stage
		Cruise missiles," observed Prahalada, director of the Defence Research and Development Laboratory two years ago, "are the present currency of power." Though in smaller denomination now, India is acquiring that currency.
		The hyperplane Avatar, the most ambitious of all, is already reaching the end of the conceptual stage and entering the planning stage.
		The kerosene-fuelled scramjet-powered vehicle is claimed to be much cheaper than the design concepts worked in the US, Germany, the UK and Japan.
	www.geocities.com /bharatvarsha1947/Feb_2003/avatar.htm (1210 words)

	Labware - MA35 Multivariable Calculus - Three Variable Calculus (Site not responding. Last check: )
		This hyperplane is defined as the tangent hyperplane to
		Note that the slices of the tangent hyperplane are the tangent planes of the slices(Why?).
		slices of the tangent hyperplane at a critical point of a function.
	www.math.gatech.edu /labware/staticlabs/32p1.html (172 words)

	Computational complexity
		The bounds derived are not applicable to a measure that, for example, uses the distances of mis-classified objects to the hyperplane.
		However, OC1 allows a certain number of adjustments to the hyperplane that do not improve the impurity, although it will never accept a change that worsens the impurity.
		Each time OC1 finds a new hyperplane that improves on the old one, it resets a counter to zero.
	www.umiacs.umd.edu /~salzberg/docs/murthy_thesis/node14.html (806 words)

	strictly separating sets
		For both 1 and 2, could you tell me whether or not there is a hyperplane that strictly separates the given sets A,B. If there is, find one.
		Projective geom, hyperplane, complex affine plane, proj closure - Projective Geometry Problem 4 Let C be the curve in a complex affine plane E. Find the infinite points of C, i.e.
		the points of the projective closure of that lie on the hyperplane at infinity...
	www.brainmass.com /homework-help/math/other/27983 (274 words)

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