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Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions
 CONSTRUCTIVE APPROXIMATION
, 1986
"... Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke. ..."
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Cited by 277 (3 self)
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Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.
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 215 (15 self)
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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 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 for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.
Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming
, 1997
"... Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by ..."
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Cited by 67 (13 self)
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Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primaldual interiorpoint algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.
Multidimensional Scaling
 Handbook of Statistics
, 2001
"... eflecting the importance or precision of dissimilarity # i j . 1. SOURCES OF DISTANCE DATA Dissimilarity information about a set of objects can arise in many different ways. We review some of the more important ones, organized by scientific discipline. 1.1. Geodesy. The most obvious application, ..."
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Cited by 33 (2 self)
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eflecting the importance or precision of dissimilarity # i j . 1. SOURCES OF DISTANCE DATA Dissimilarity information about a set of objects can arise in many different ways. We review some of the more important ones, organized by scientific discipline. 1.1. Geodesy. The most obvious application, perhaps, is in sciences in which distance is measured directly, although generally with error. This happens, for instance, in triangulation in geodesy. We have measurements which are approximately equal to distances, either Euclidean or spherical, depending on the scale of the experiment. In other examples, measured distances are less directly related to physical distances. For example, we could measure airplane or road or train travel distances between different cities. Physical distance is usually not the only factor determining these types of dissimilarities. 1 2 J. DE LEEUW<
Geometric Methods for Feature Extraction and Dimensional Reduction
 In L. Rokach and O. Maimon (Eds.), Data
, 2005
"... Abstract We give a tutorial overview of several geometric methods for feature extraction and dimensional reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component anal ..."
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Cited by 31 (1 self)
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Abstract We give a tutorial overview of several geometric methods for feature extraction and dimensional reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component analysis (PCA), kernel PCA, probabilistic PCA, and oriented PCA; and for the manifold methods, we review multidimensional scaling (MDS), landmark MDS, Isomap, locally linear embedding, Laplacian eigenmaps and spectral clustering. The Nyström method, which links several of the algorithms, is also reviewed. The goal is to provide a selfcontained review of the concepts and mathematics underlying these algorithms.
Properties of Euclidean and nonEuclidean distance matrices, Linear Algebra Appl
, 1985
"... A distance matrix D of order n is symmetric with elements idfj, where d,, = 0. D is Euclidean when the in(n 1) quantities dij can be generated as the distances between a set of n points, X (n X p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p = r ..."
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Cited by 26 (0 self)
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A distance matrix D of order n is symmetric with elements idfj, where d,, = 0. D is Euclidean when the in(n 1) quantities dij can be generated as the distances between a set of n points, X (n X p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p = rank(X) of any generating X; in general p + 1 and p +2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when p = rank(D) it is shown that, depending on whether erDe is not or is zero, the generating points lie in either p = p 1 dimensions, in which case they lie on a hypersphere, or in p = p 2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is nonEuclidean its dimensionality p = r + s will comprise r real and s imaginary columns of X, and (T, s) are invariant for all generating X of minimal rank.
Molecular distance geometry methods: From continuous to discrete
, 2009
"... Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context ..."
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Cited by 18 (18 self)
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Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context of molecular distance geometry, the distances are usually known because of chemical properties and Nuclear Magnetic Resonance experiments; sensor networks can estimate their relative distance by recording the power loss during a twoway exchange; finally, when drawing graphs in 2D or 3D, the graph to be drawn is given, and therefore distances between vertices can be computed. Distance geometry problems involve a search in a continuous Euclidean space, but sometimes the problem structure helps reduce the search to a discrete set of points. In this paper we survey some continuous and discrete methods for solving some problems of molecular distance geometry. 1
Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming
 Linear Algebra Appl
, 2005
"... Abstract A partial predistance matrix A is a matrix with zero diagonal and with certain elements fixed to given nonnegative values; the other elements are considered free. The Euclidean distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a Euclidea ..."
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Cited by 14 (6 self)
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Abstract A partial predistance matrix A is a matrix with zero diagonal and with certain elements fixed to given nonnegative values; the other elements are considered free. The Euclidean distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a Euclidean distance matrix, EDM. The nearest (or approximate) Euclidean distance matrix problem is to find a Euclidean
Sensor Network Localization, Euclidean Distance Matrix. Completions, and Graph Realization
, 2008
"... ..."
On the Embeddability of Weighted Graphs in Euclidean Spaces
, 1998
"... Given an incomplete edgeweighted graph, G = (V; E; !), G is said to be embeddable in ! r , or rembeddable, if the vertices of G can be mapped to points in ! r such that every two adjacent vertices v i , v j of G are mapped to points x i , x j 2 ! r whose Euclidean distance is equal to t ..."
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Cited by 11 (1 self)
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Given an incomplete edgeweighted graph, G = (V; E; !), G is said to be embeddable in ! r , or rembeddable, if the vertices of G can be mapped to points in ! r such that every two adjacent vertices v i , v j of G are mapped to points x i , x j 2 ! r whose Euclidean distance is equal to the weight of the edge (v i ; v j ). Barvinok [3] proved that if G is rembeddable for some r, then it is r embeddable where r = b( p 8jEj + 1 \Gamma 1)=2c. In this paper we provide a constructive proof of this result by presenting an algorithm to construct such an r embedding. 1 Introduction Let G = (V; E;!) be an incomplete undirected edgeweighted graph with vertex set V = fv 1 ; v 2 ; : : : ; v n g, edge set E ae V \Theta V and a nonnegative weight Email aalfakih@orion.math.uwaterloo.ca y Research supported by Natural Sciences Engineering Research Council Canada. Email henry@orion.math.uwaterloo.ca. ! ij for each (v i ; v j ) 2 E. G is said to be rembeddable if th...