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## Geometria de distâncias euclidianas e aplicações; Euclidean distance geometry and applications

Jorge Ferreira Alencar Lima
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 23/01/2015 PT
Relevância na Pesquisa
76.24%

## Euclidean distance matrices: special subsets, systems of coordinates and multibalanced matrices

Tarazaga,Pablo; Sterba-Boatwright,Blair; Wijewardena,Kithsiri
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2007 EN
Relevância na Pesquisa
86.26%
In this paper we present special subsets of positive semidefinite matrices where the linear function k becomes a geometric similarity and its inverse can be easily computed. The images of these special subsets are characterized geometrically. We also study the systems of coordinates for spherical matrices and at the end, we introduce the class of multibalanced distance matrices.

## On the eigenvalues of Euclidean distance matrices

Alfakih,A.Y.
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2008 EN
Relevância na Pesquisa
96.25%
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 polynomials 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.

## On the Nonnegative Rank of Euclidean Distance Matrices

Lin, Matthew M.; Chu, Moody T.
Fonte: PubMed Publicador: PubMed
Tipo: Artigo de Revista Científica
EN
Relevância na Pesquisa
56%
The Euclidean distance matrix for n distinct points in ℝr is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.

## Positive semidefinite metric learning using boosting-like algorithms

Shen, C.; Kim, J.; Wang, L.; Van Den Hengel, A.
Fonte: MIT Press Publicador: MIT Press
Tipo: Artigo de Revista Científica
Publicado em //2012 EN
Relevância na Pesquisa
36.07%
The success of many machine learning and pattern recognition methods relies heavily upon the identification of on an appropriate distance metric on the input data. It is often beneficial to learn such a metric from the input training data, instead of using a default one such as the Euclidean distance. In this work, we propose a boosting-based technique, termed BoostMetric, for learning a quadratic Mahalanobis distance metric. Learning a valid Mahalanobis distance metric requires enforcing the constraint that the matrix parameter to the metric remains positive semidefinite. Semidefinite programming is often used to enforce this constraint, but does not scale well and not easy to implement. BoostMetric is instead based on the observation that any positive semidefinite matrix can be decomposed into a linear combination of trace-one rank-one matrices. BoostMetric thus uses rank-one positive semidefinite matrices as weak learners within an efficient and scalable boosting-based learning process. The resulting methods are easy to implement, efficient, and can accommodate various types of constraints. We extend traditional boosting algorithms in that its weak learner is a positive semidefinite matrix with trace and rank being one rather than a classifier or regressor. Experiments on various datasets demonstrate that the proposed algorithms compare favorably to those state-of-the-art methods in terms of classification accuracy and running time.; Chunhua Shen...

## Representing functional data in reproducing Kernel Hilbert spaces with applications to clustering, classification and time series problems

González Hernández, Javier
Tipo: info:eu-repo/semantics/doctoralThesis; info:eu-repo/semantics/doctoralThesis Formato: application/octet-stream; application/octet-stream; application/pdf
ENG
Relevância na Pesquisa
36.27%
In modern data analysis areas such as Image Analysis, Chemometrics or Information Retrieval the raw data are often complex and their representation in Euclidean spaces is not straightforward. However most statistical data analysis techniques are designed to deal with points in Euclidean spaces and hence a representation of the data in some Euclidean coordinate system is always required as a previous step to apply multivariate analysis techniques. This process is crucial to guarantee the success of the data analysis methodologies and will be a core contribution of this thesis. In this work we will develop general data representation techniques in the framework of Functional Data Analysis (FDA) for classification and clustering problems. In Chapter 1 we motivate the problems to solve, describe the roadmap of the contributions and set up the notation of this work. In Chapter 2 we review some aspects concerning Reproducing Kernel Hilbert Spaces (RKHSs), Regularization Theory Integral Operators, Support Vector Machines and Kernel Combinations. In Chapter 3 we propose a new methodology to obtain finite-dimensional representations of functional data. The key idea is to consider each functional curve as a point in a general function space and then project these points onto a Reproducing Kernel Hilbert Space (RKHS) with the aid of Regularization theory. We will describe the projection methods...

## Characterizing graphs with convex and connected configuration spaces

Sitharam, Meera; Gao, Heping
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36%
We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS), which includes Linkages and Frameworks. Each representation corresponds to a choice of Cayley parameters and yields a different parametrized configuration space. Significantly, we give purely graph-theoretic, forbidden minor characterizations that capture (i) the class of graphs that always admit efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. In addition, our results are tight: we show counterexamples to obvious extensions. This is the first step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. We discuss several future theoretical and applied research directions. Some of our proofs employ an unusual interplay of (a) classical analytic results related to positive semi-definiteness of Euclidean distance matrices, with (b) recent forbidden minor characterizations and algorithms related to the notion of d-realizability of EDCS. We further introduce a novel type of restricted edge contraction or reduction to a graph minor, a "trick" that we anticipate will be useful in other situations.

## Spectra of Euclidean Random Matrices

Mezard, M.; Parisi, G.; Zee, A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We study the spectrum of a random matrix, whose elements depend on the Euclidean distance between points randomly distributed in space. This problem is widely studied in the context of the Instantaneous Normal Modes of fluids and is particularly relevant at the glass transition. We introduce a systematic study of this problem through its representation by a field theory. In this way we can easily construct a high density expansion, which can be resummed producing an approximation to the spectrum similar to the Coherent Potential Approximation for disordered systems.; Comment: 10 pages, 4 figures

## On stochastic generation of ultrametrics in high-dimension Euclidean spaces

Zubarev, Alexander P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.12%
The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the ultrametric is presented. The values of the ultrametric distance matrix is completely determined by variances of point coordinates. Probabilistic algorithm for the generation of finite ultrametric structures of any topology in high-dimensional Euclidean space is presented. The validity of the algorithm is demonstrated by explicit calculations of distance matrices with fixed dimensions and ultrametricity indexes for various dimensions of Euclidean space.; Comment: 13 pages, 2 figures

## Distance matrices and isometric embeddings

Bogomolny, E.; Bohigas, O.; Schmit, C.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.08%
We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenvalues except one non-negative. Several generalizations are discussed.; Comment: 17 pages

## Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction

Ding, Chao; Qi, Hou-Duo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.06%
Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP) to reconstruct such faithful representations. While those SDP models are capable of producing high quality configuration numerically, they suffer two major drawbacks. One is that there exist no theoretically guaranteed bounds on the quality of the configuration. The other is that they are slow in computation when the data points are beyond moderate size. In this paper, we propose a convex optimization model of Euclidean distance matrices. We establish a non-asymptotic error bound for the random graph model with sub-Gaussian noise, and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low-rank matrix up to a logarithmic factor. Our results partially explain why MVU and MVE often work well. Moreover, we develop a fast inexact accelerated proximal gradient method. Numerical experiments show that the model can produce configurations of high quality on large data points that the SDP approach would struggle to cope with.; Comment: 44 pages...

## Universal rigidity of bar frameworks in general position: a Euclidean distance matrix approach

Alfakih, A. Y.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.31%
A configuration p in r-dimensional Euclidean space is a finite collection of labeled points p^1,p^2,...,p^n in R^r that affinely span R^r. Each configuration p defines a Euclidean distance matrix D_p = (d_ij) = (||p^i-p^j||^2), where ||.|| denotes the Euclidean norm. A fundamental problem in distance geometry is to find out whether or not, a given proper subset of the entries of D_p suffices to uniquely determine the entire matrix D_p. This problem is known as the universal rigidity problem of bar frameworks. In this chapter, we present a unified approach for the universal rigidity of bar frameworks, based on Euclidean distance matrices (EDMs), or equivalently, on projected Gram matrices. This approach makes the universal rigidity problem amenable to semi-definite programming methodology. Using this approach, we survey some recently obtained results and their proofs, emphasizing the case where the points p^1,...,p^n are in general position.; Comment: To appear in "Distance geometry: with applications to molecular conformation and sensor networks" edited by Mucherino et al. arXiv admin note: text overlap with arXiv:1009.3318

## Coordinate shadows of semi-definite and Euclidean distance matrices

Drusvyatskiy, D.; Pataki, G.; Wolkowicz, H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.1%
We consider the projected semi-definite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semi-definite and Euclidean distance completion problems. We classify when these sets are closed, and use the boundary structure of these two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the "minimal cones" of these problems admit combinatorial characterizations. As a byproduct, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping.; Comment: 21 pages

## On the Geometric Interpretation of the Nonnegative Rank

Gillis, Nicolas; Glineur, François
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.15%
The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.

## Counting real critical points of the distance to orthogonally invariant matrix sets

Drusvyatskiy, Dmitriy; Lee, Hon-Leung; Thomas, Rekha R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.14%
Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on "transfer principles" that allow calculations to be done in the space of singular values of the matrices in the orthogonally invariant set. The calculations often simplify greatly and yield transparent formulas. We illustrate the method on several examples, and compare our results to the recently introduced notion of Euclidean distance degree of an algebraic variety.; Comment: 21 pages, 7 figures; a part of Section 5 in v1 became Appendix A

## The Euclidean distance degree of an algebraic variety

Draisma, Jan; Horobet, Emil; Ottaviani, Giorgio; Sturmfels, Bernd; Thomas, Rekha R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.13%
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.; Comment: to appear in Foundations of Computational Mathematics

## Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

Dokmanic, Ivan; Parhizkar, Reza; Ranieri, Juri; Vetterli, Martin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.19%
Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.; Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revision

## On the Coherence Properties of Random Euclidean Distance Matrices

Kalogerias, Dionysios S.; Petropulu, Athina P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.1%
In the present paper we focus on the coherence properties of general random Euclidean distance matrices, which are very closely related to the respective matrix completion problem. This problem is of great interest in several applications such as node localization in sensor networks with limited connectivity. Our results can directly provide the sufficient conditions under which an EDM can be successfully recovered with high probability from a limited number of measurements.; Comment: 5 pages, SPAWC 2013

## Riemannian Metric Learning for Symmetric Positive Definite Matrices

Vemulapalli, Raviteja; Jacobs, David W.
Tipo: Artigo de Revista Científica
In Micchelli's paper "Interpolation of scattered data: distance matrices and conditionally positive functions", deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated by $p$-norms. In particular, we show that, for any $p\in (1, 2)$, and for distinct points $x^1, ..., x^n \in {\cal R}^d$, where $n$ and $d$ may be any positive integers, with the proviso that $n \ge 2$, the matrix $A \in {\cal R}^{n \times n}$ defined by $$A_{ij} = \Vert x^i - x^j \Vert_p , \hbox{ for } 1 \le i, j \le n,$$ satisfies $$(-1)^{n-1}\det A > 0 .$$ We also show how to construct, for every $p > 2$, a configuration of distinct points in some ${\cal R}^d$ giving a singular $p$-norm distance matrix. Thus radial basis function interpolation using $p$-norms is uniquely determined by any distinct data for $p \in (1,2]$, but not so for $p > 2$.