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Vector subspace
, 2013
"... Subspace spanned by a set of vectors Null space and column space of a matrix Kernel and range of a linear transformation Linearly independent sets and bases Bases for Nul{A} and Col{A} ..."
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Subspace spanned by a set of vectors Null space and column space of a matrix Kernel and range of a linear transformation Linearly independent sets and bases Bases for Nul{A} and Col{A}
Ultrabarrelled spaces and dense vector subspaces
, 1990
"... Let {E.x) be a Hausdorff topological vector space (tvs) over the field K € CRC). Let M be a vector subspace of the algebraic dual E * of E. If %{x) is a 0basis for T, then the sets ..."
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Let {E.x) be a Hausdorff topological vector space (tvs) over the field K € CRC). Let M be a vector subspace of the algebraic dual E * of E. If %{x) is a 0basis for T, then the sets
THE MAXLENGTHVECTOR LINE OF BEST FIT TO A COLLECTION OF VECTOR SUBSPACES
"... (Communicated by) Abstract. Let C = {V1, V2,..., Vk} be a finite collection of nontrivial subspaces of a finite dimensional real vector space V. Let L denote a one dimensional subspace of V and let θ(L, Vi) denote the principal (or canonical) angle between L and Vi. We are interested in finding al ..."
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(Communicated by) Abstract. Let C = {V1, V2,..., Vk} be a finite collection of nontrivial subspaces of a finite dimensional real vector space V. Let L denote a one dimensional subspace of V and let θ(L, Vi) denote the principal (or canonical) angle between L and Vi. We are interested in finding
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
 SIAM J. SCI. STAT. COMPUT
, 1986
"... We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2orthogonal basis of Krylov subspaces. It can be considered a ..."
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Cited by 2073 (40 self)
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We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2orthogonal basis of Krylov subspaces. It can be considered
Probabilistic Principal Component Analysis
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B
, 1999
"... Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximumlikelihood estimation of paramet ..."
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Cited by 709 (5 self)
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Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximumlikelihood estimation
Reversible jump Markov chain Monte Carlo computation and Bayesian model determination
 Biometrika
, 1995
"... Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some xed standard underlying measure. They have therefore not been available for application to Bayesian model determi ..."
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Cited by 1342 (23 self)
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determination, where the dimensionality of the parameter vector is typically not xed. This article proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of di ering dimensionality, which is exible and entirely constructive. It should therefore
Using Linear Algebra for Intelligent Information Retrieval
 SIAM REVIEW
, 1995
"... Currently, most approaches to retrieving textual materials from scientific databases depend on a lexical match between words in users' requests and those in or assigned to documents in a database. Because of the tremendous diversity in the words people use to describe the same document, lexical ..."
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Cited by 675 (18 self)
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by 200300 of the largest singular vectors are then matched against user queries. We call this retrieval method Latent Semantic Indexing (LSI) because the subspace represents important associative relationships between terms and documents that are not evident in individual documents. LSI is a completely
Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization
, 2007
"... The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative ..."
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Cited by 561 (20 self)
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, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NPhard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds
From Few to many: Illumination cone models for face recognition under variable lighting and pose
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... We present a generative appearancebased method for recognizing human faces under variation in lighting and viewpoint. Our method exploits the fact that the set of images of an object in fixed pose, but under all possible illumination conditions, is a convex cone in the space of images. Using a smal ..."
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Cited by 755 (12 self)
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conditions. The pose space is then sampled, and for each pose the corresponding illumination cone is approximated by a lowdimensional linear subspace whose basis vectors are estimated using the generative model. Our recognition algorithm assigns to a test image the identity of the closest approximated
A Linear Classification Of 6Element Sequences Of Vector Subspaces In The 4Dimensional Vector Space
"... this paper is a criterion of GL(V ) congruence for two polyhedra (in the vector space V ) of a family A with the assumption that the planes belonging to these polyhedra are given by linear independent systems of generators. By a polyhedron in a vector space we mean a finite sequence of subspaces in ..."
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this paper is a criterion of GL(V ) congruence for two polyhedra (in the vector space V ) of a family A with the assumption that the planes belonging to these polyhedra are given by linear independent systems of generators. By a polyhedron in a vector space we mean a finite sequence of subspaces
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