Euclidean Distance Matrix Trick
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Eigenvalues of Euclidean Distance Matrices and Rs-Majorization on R2
Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 1{4 Eigenvalues of Euclidean Distance Matrices and rs-majorization on R2 Asma Ilkhanizadeh Manesh∗ Department of Pure Mathematics, Vali-e-Asr University of Rafsanjan Alemeh Sheikh Hoseini Department of Pure Mathematics, Shahid Bahonar University of Kerman Abstract Let D1 and D2 be two Euclidean distance matrices (EDMs) with correspond- ing positive semidefinite matrices B1 and B2 respectively. Suppose that λ(A) = ((λ(A)) )n is the vector of eigenvalues of a matrix A such that (λ(A)) ... i i=1 1 ≥ ≥ (λ(A))n. In this paper, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs, on R will be investigated. ≺ Keywords: Euclidean distance matrices, Rs-majorization. Mathematics Subject Classification [2010]: 34B15, 76A10 1 Introduction An n n nonnegative and symmetric matrix D = (d2 ) with zero diagonal elements is × ij called a predistance matrix. A predistance matrix D is called Euclidean or a Euclidean distance matrix (EDM) if there exist a positive integer r and a set of n points p1, . , pn r 2 2 { } such that p1, . , pn R and d = pi pj (i, j = 1, . , n), where . denotes the ∈ ij k − k k k usual Euclidean norm. The smallest value of r that satisfies the above condition is called the embedding dimension. As is well known, a predistance matrix D is Euclidean if and 1 1 t only if the matrix B = − P DP with P = I ee , where I is the n n identity matrix, 2 n − n n × and e is the vector of all ones, is positive semidefinite matrix. -
Polynomial Approximation Algorithms for Belief Matrix Maintenance in Identity Management
Polynomial Approximation Algorithms for Belief Matrix Maintenance in Identity Management Hamsa Balakrishnan, Inseok Hwang, Claire J. Tomlin Dept. of Aeronautics and Astronautics, Stanford University, CA 94305 hamsa,ishwang,[email protected] Abstract— Updating probabilistic belief matrices as new might be constrained to some prespecified (but not doubly- observations arrive, in the presence of noise, is a critical part stochastic) row and column sums. This paper addresses the of many algorithms for target tracking in sensor networks. problem of updating belief matrices by scaling in the face These updates have to be carried out while preserving sum constraints, arising for example, from probabilities. This paper of uncertainty in the system and the observations. addresses the problem of updating belief matrices to satisfy For example, consider the case of the belief matrix for a sum constraints using scaling algorithms. We show that the system with three objects (labelled 1, 2 and 3). Suppose convergence behavior of the Sinkhorn scaling process, used that, at some instant, we are unsure about their identities for scaling belief matrices, can vary dramatically depending (tagged X, Y and Z) completely, and our belief matrix is on whether the prior unscaled matrix is exactly scalable or only almost scalable. We give an efficient polynomial-time algo- a 3 × 3 matrix with every element equal to 1/3. Let us rithm based on the maximum-flow algorithm that determines suppose the we receive additional information that object whether a given matrix is exactly scalable, thus determining 3 is definitely Z. Then our prior, but constraint violating the convergence properties of the Sinkhorn scaling process. -
Tight Frames and Their Symmetries
Technical Report 9 December 2003 Tight Frames and their Symmetries Richard Vale, Shayne Waldron Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: [email protected] (http:www.math.auckland.ac.nz/˜waldron) e–mail: [email protected] ABSTRACT The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries. Key Words: Tight frames, isometric tight frames, Gram matrix, multivariate orthogonal polynomials, symmetry groups, harmonic frames, representation theory, wavelets AMS (MOS) Subject Classifications: primary 05B20, 33C50, 20C15, 42C15, sec- ondary 52B15, 42C40 0 1. Introduction u1 u2 u3 2 The three equally spaced unit vectors u1, u2, u3 in IR provide the following redundant representation 2 3 f = f, u u , f IR2, (1.1) 3 h ji j ∀ ∈ j=1 X which is the simplest example of a tight frame. Such representations arose in the study of nonharmonic Fourier series in L2(IR) (see Duffin and Schaeffer [DS52]) and have recently been used extensively in the theory of wavelets (see, e.g., Daubechies [D92]). -
On the Eigenvalues of Euclidean Distance Matrices
“main” — 2008/10/13 — 23:12 — page 237 — #1 Volume 27, N. 3, pp. 237–250, 2008 Copyright © 2008 SBMAC ISSN 0101-8205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH∗ Department of Mathematics and Statistics University of Windsor, Windsor, Ontario N9B 3P4, Canada E-mail: [email protected] Abstract. In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic poly- nomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues. Mathematical subject classification: 51K05, 15A18, 05C50. Key words: Euclidean distance matrices, eigenvalues, equitable partitions, characteristic poly- nomial. 1 Introduction ( ) An n ×n nonzero matrix D = di j is called a Euclidean distance matrix (EDM) 1, 2,..., n r if there exist points p p p in some Euclidean space < such that i j 2 , ,..., , di j = ||p − p || for all i j = 1 n where || || denotes the Euclidean norm. i , ,..., Let p , i ∈ N = {1 2 n}, be the set of points that generate an EDM π π ( , ,..., ) D. An m-partition of D is an ordered sequence = N1 N2 Nm of ,..., nonempty disjoint subsets of N whose union is N. The subsets N1 Nm are called the cells of the partition. The n-partition of D where each cell consists #760/08. Received: 07/IV/08. Accepted: 17/VI/08. ∗Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS. -
Smith Normal Formal of Distance Matrix of Block Graphs∗†
Ann. of Appl. Math. 32:1(2016); 20-29 SMITH NORMAL FORMAL OF DISTANCE MATRIX OF BLOCK GRAPHS∗y Jing Chen1;2,z Yaoping Hou2 (1. The Center of Discrete Math., Fuzhou University, Fujian 350003, PR China; 2. School of Math., Hunan First Normal University, Hunan 410205, PR China) Abstract A connected graph, whose blocks are all cliques (of possibly varying sizes), is called a block graph. Let D(G) be its distance matrix. In this note, we prove that the Smith normal form of D(G) is independent of the interconnection way of blocks and give an explicit expression for the Smith normal form in the case that all cliques have the same size, which generalize the results on determinants. Keywords block graph; distance matrix; Smith normal form 2000 Mathematics Subject Classification 05C50 1 Introduction Let G be a connected graph (or strong connected digraph) with vertex set f1; 2; ··· ; ng. The distance matrix D(G) is an n × n matrix in which di;j = d(i; j) denotes the distance from vertex i to vertex j: Like the adjacency matrix and Lapla- cian matrix of a graph, D(G) is also an integer matrix and there are many results on distance matrices and their applications. For distance matrices, Graham and Pollack [10] proved a remarkable result that gives a formula of the determinant of the distance matrix of a tree depend- ing only on the number n of vertices of the tree. The determinant is given by det D = (−1)n−1(n − 1)2n−2: This result has attracted much interest in algebraic graph theory. -
Week 8-9. Inner Product Spaces. (Revised Version) Section 3.1 Dot Product As an Inner Product
Math 2051 W2008 Margo Kondratieva Week 8-9. Inner product spaces. (revised version) Section 3.1 Dot product as an inner product. Consider a linear (vector) space V . (Let us restrict ourselves to only real spaces that is we will not deal with complex numbers and vectors.) De¯nition 1. An inner product on V is a function which assigns a real number, denoted by < ~u;~v> to every pair of vectors ~u;~v 2 V such that (1) < ~u;~v>=< ~v; ~u> for all ~u;~v 2 V ; (2) < ~u + ~v; ~w>=< ~u;~w> + < ~v; ~w> for all ~u;~v; ~w 2 V ; (3) < k~u;~v>= k < ~u;~v> for any k 2 R and ~u;~v 2 V . (4) < ~v;~v>¸ 0 for all ~v 2 V , and < ~v;~v>= 0 only for ~v = ~0. De¯nition 2. Inner product space is a vector space equipped with an inner product. Pn It is straightforward to check that the dot product introduces by ~u ¢ ~v = j=1 ujvj is an inner product. You are advised to verify all the properties listed in the de¯nition, as an exercise. The dot product is also called Euclidian inner product. De¯nition 3. Euclidian vector space is Rn equipped with Euclidian inner product < ~u;~v>= ~u¢~v. De¯nition 4. A square matrix A is called positive de¯nite if ~vT A~v> 0 for any vector ~v 6= ~0. · ¸ 2 0 Problem 1. Show that is positive de¯nite. 0 3 Solution: Take ~v = (x; y)T . Then ~vT A~v = 2x2 + 3y2 > 0 for (x; y) 6= (0; 0). -
On Asymmetric Distances
Analysis and Geometry in Metric Spaces Research Article • DOI: 10.2478/agms-2013-0004 • AGMS • 2013 • 200-231 On Asymmetric Distances Abstract ∗ In this paper we discuss asymmetric length structures and Andrea C. G. Mennucci asymmetric metric spaces. A length structure induces a (semi)distance function; by using Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempo- tent. As a typical application, we consider the length of paths Received 24 March 2013 defined by a Finslerian functional in Calculus of Variations. Accepted 15 May 2013 In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space. Keywords Asymmetric metric • general metric • quasi metric • ostensible metric • Finsler metric • run–continuity • intrinsic metric • path metric • length structure MSC: 54C99, 54E25, 26A45 © Versita sp. z o.o. 1. Introduction Besides, one insists that the distance function be symmetric, that is, d(x; y) = d(y; x) (This unpleasantly limits many applications [...]) M. Gromov ([12], Intr.) The main purpose of this paper is to study an asymmetric metric theory; this theory naturally generalizes the metric part of Finsler Geometry, much as symmetric metric theory generalizes the metric part of Riemannian Geometry. -
Gram Matrix and Orthogonality in Frames 1
U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 1, 2018 ISSN 1223-7027 GRAM MATRIX AND ORTHOGONALITY IN FRAMES Abolhassan FEREYDOONI1 and Elnaz OSGOOEI 2 In this paper, we aim at introducing a criterion that determines if f figi2I is a Bessel sequence, a frame or a Riesz sequence or not any of these, based on the norms and the inner products of the elements in f figi2I. In the cases of Riesz and Bessel sequences, we introduced a criterion but in the case of a frame, we did not find any answers. This criterion will be shown by K(f figi2I). Using the criterion introduced, some interesting extensions of orthogonality will be presented. Keywords: Frames, Bessel sequences, Orthonormal basis, Riesz bases, Gram matrix MSC2010: Primary 42C15, 47A05. 1. Preliminaries Frames are generalizations of orthonormal bases, but, more than orthonormal bases, they have shown their ability and stability in the representation of functions [1, 4, 10, 11]. The frames have been deeply studied from an abstract point of view. The results of such studies have been used in concrete frames such as Gabor and Wavelet frames which are very important from a practical point of view [2, 9, 5, 8]. An orthonormal set feng in a Hilbert space is characterized by a simple relation hem;eni = dm;n: In the other words, the Gram matrix is the identity matrix. Moreover, feng is an orthonor- mal basis if spanfeng = H. But, for frames the situation is more complicated; i.e., the Gram Matrix has no such a simple form. -
Uniqueness of Low-Rank Matrix Completion by Rigidity Theory
UNIQUENESS OF LOW-RANK MATRIX COMPLETION BY RIGIDITY THEORY AMIT SINGER∗ AND MIHAI CUCURINGU† Abstract. The problem of completing a low-rank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constructing efficient algorithms for exact or approximate recovery of the missing matrix entries and proving lower bounds for the number of known entries that guarantee a successful recovery with high probability. A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances. Rigidity theory answers basic questions regarding the uniqueness of the realization satisfying a given partial set of distances. We observe that basic ideas and tools of rigidity theory can be adapted to determine uniqueness of low-rank matrix completion, where inner products play the role that distances play in rigidity theory. This observation leads to efficient randomized algorithms for testing necessary and sufficient conditions for local completion and for testing sufficient conditions for global completion. Crucial to our analysis is a new matrix, which we call the completion matrix, that serves as the analogue of the rigidity matrix. Key words. Low rank matrices, missing values, rigidity theory, iterative methods, collaborative filtering. AMS subject classifications. 05C10, 05C75, 15A48 1. Introduction. Can the missing entries of an incomplete real valued matrix be recovered? Clearly, a matrix can be completed in an infinite number of ways by replacing the missing entries with arbitrary values. -
Inner Products and Norms (Part III)
Inner Products and Norms (part III) Prof. Dan A. Simovici UMB 1 / 74 Outline 1 Approximating Subspaces 2 Gram Matrices 3 The Gram-Schmidt Orthogonalization Algorithm 4 QR Decomposition of Matrices 5 Gram-Schmidt Algorithm in R 2 / 74 Approximating Subspaces Definition A subspace T of a inner product linear space is an approximating subspace if for every x 2 L there is a unique element in T that is closest to x. Theorem Let T be a subspace in the inner product space L. If x 2 L and t 2 T , then x − t 2 T ? if and only if t is the unique element of T closest to x. 3 / 74 Approximating Subspaces Proof Suppose that x − t 2 T ?. Then, for any u 2 T we have k x − u k2=k (x − t) + (t − u) k2=k x − t k2 + k t − u k2; by observing that x − t 2 T ? and t − u 2 T and applying Pythagora's 2 2 Theorem to x − t and t − u. Therefore, we have k x − u k >k x − t k , so t is the unique element of T closest to x. 4 / 74 Approximating Subspaces Proof (cont'd) Conversely, suppose that t is the unique element of T closest to x and x − t 62 T ?, that is, there exists u 2 T such that (x − t; u) 6= 0. This implies, of course, that u 6= 0L. We have k x − (t + au) k2=k x − t − au k2=k x − t k2 −2(x − t; au) + jaj2 k u k2 : 2 2 Since k x − (t + au) k >k x − t k (by the definition of t), we have 2 2 1 −2(x − t; au) + jaj k u k > 0 for every a 2 F. -
Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. -
Nonlinear Mapping and Distance Geometry
Nonlinear Mapping and Distance Geometry Alain Franc, Pierre Blanchard , Olivier Coulaud arXiv:1810.08661v2 [cs.CG] 9 May 2019 RESEARCH REPORT N° 9210 October 2018 Project-Teams Pleiade & ISSN 0249-6399 ISRN INRIA/RR--9210--FR+ENG HiePACS Nonlinear Mapping and Distance Geometry Alain Franc ∗†, Pierre Blanchard ∗ ‡, Olivier Coulaud ‡ Project-Teams Pleiade & HiePACS Research Report n° 9210 — version 2 — initial version October 2018 — revised version April 2019 — 14 pages Abstract: Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: Distance Geometry Problem is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear Mapping is a weighted Least Square Scaling (LSS) method. We show how all these methods (LSS, NLM, DGP) can be assembled in a common framework, being each identified as an instance of an optimization problem with a choice of a weight matrix. We study the continuity between the solutions (which are point clouds) when the weight matrix varies, and the compactness of the set of solutions (after centering). We finally study a numerical example, showing that solving the optimization problem is far from being simple and that the numerical solution for a given procedure may be trapped in a local minimum. Key-words: Distance Geometry; Nonlinear Mapping, Discrete Metric Space, Least Square Scaling; optimization ∗ BIOGECO, INRA, Univ. Bordeaux, 33610 Cestas, France † Pleiade team - INRIA Bordeaux-Sud-Ouest, France ‡ HiePACS team, Inria Bordeaux-Sud-Ouest,