The embedding problem for predistance matrices

Glunt, W.; Hayden, T. L.; Liu, Weimin

doi:10.1007/bf02461553

Cited by 32 publications

(8 citation statements)

References 25 publications

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“…Nevertheless, as reported in Browne (1987), the algorithms based on this approach produce more accurate results. The same was reported in Glunt et al (1991) and their related papers. It seems that the most powerful algorithm based on this MDS approach is the one developed by Kearsley et al (1998).…”

supporting

confidence: 81%

DINDSCAL: direct INDSCAL

Trendafilov

2011

Stat Comput

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The well-known INDSCAL model for simultaneous metric multidimensional scaling (MDS) of three-way data analyzes doubly centered matrices of squared dissimilarities. An alternative approach, called for short DIND-SCAL, is proposed for analyzing directly the input matrices of squared dissimilarities. An important consequence is that missing values can be easily handled. The DIND-SCAL problem is solved by means of the projected gradient approach. First, the problem is transformed into a gradient dynamical system on a product matrix manifold (of Stiefel sub-manifold of zero-sum matrices and non-negative diagonal matrices). The constructed dynamical system can be numerically integrated which gives a globally convergent algorithm for solving the DINDSCAL. The DINDSCAL problem and its solution are illustrated by well-known data routinely used in metric MDS and INDSCAL. Alternatively, the problem can also be solved by iterative algorithm based on the conjugate (projected) gradient method, which MATLAB implementation is enclosed as an appendix.

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supporting

confidence: 81%

DINDSCAL: direct INDSCAL

Trendafilov

2011

Stat Comput

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“…For the case when r = n -1, Gaffke and Mathar [9] and Glunt et al [11] developed algorithms which yield a global minimum of the Sstress function. Recently, Alfakih et al [1] have also considered to minimize the same error function by solving a positive semidefinite programming problem with a quadratic objective function.…”

Section: (Ij)esmentioning

confidence: 99%

“…Also, smoothing techniques by Moré and Wu [18] and Stochastic/perturbation approaches by Zou et al [25] have been applied to the S-stress function. We note that the differentiability of the S-stress function brings rich mathematical properties such as those developed in the literature [9,11,1]. It has been shown that the minimization of the S-stress function can be formulated as a convex minimization problem when r = n -1.…”

Section: (Ij)esmentioning

confidence: 99%

“…We will show that the similar properties discussed in [11,1] will be exploited to develop a (linear) positive semidefinite programming problem (SDP) for obtaining a global minimizer of the absolute differences when r = n-1. It should be emphasized that the computational advantage of our method has been brought out by the recent development of many efficient algorithms for SDP [20,15,24,2], which enables us to solve fairly large scale problems in a practical amount of time.…”

Section: (Ij)esmentioning

confidence: 99%

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Positive semidefinite relaxations for distance geometry problems

Yajima

2002

Japan J. Indust. Appl. Math.

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We consider distance geometry problems for determining r-dimensional coordinates of n points from a given set of the distances between the points. This problem is a fundamental problem in molecular biology for finding the structure of proteins from NMR (Nuclear Magnetic Resonance) data.We formulate the problem as the minimization of an error function defined by the sum of the absolute differences, which is a typical nonconvex optimization problem. We show that this problem can be reduced into a concave quadratic minimization problem. We propose positive semidefinite relaxations as well as local search procedures for this problem. Our numerical results show that we can generate acceptable solutions of the problem with up to 800 points.

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“…Browne proposed a method for computing s-stress based on Newton-Raphson root finding [47]. Glunt reports that the method by Browne converges to the global minimum of (25) in 90% of the test cases in his dataset 6 [48].…”

mentioning

confidence: 99%

Euclidean Distance Matrices: Essential theory, algorithms, and applications

Dokmanić

Parhizkar²,

Ranieri

et al. 2015

IEEE Signal Process. Mag.

468

276

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Euclidean distance matrices (EDMs) are matrices of the 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, and show how the 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. The code for all of the described algorithms and to generate the figures in the article is available online at http://lcav.epfl.ch/ivan.dokmanic. Finally, we suggest directions for further research. IntroductIonImagine that you land at Geneva International Airport with the Swiss train schedule but no map. Perhaps surprisingly, this may be sufficient to reconstruct a rough (or not so rough) map of the Alpine country, even if the train times poorly translate to distances or if some of the times are unknown. The way to do it is by using EDMs; for an example, see "Swiss Trains (Swiss Map Reconstruction)."We often work with distances because they are convenient to measure or estimate. In wireless sensor networks, for example, the sensor nodes measure the received signal strengths of the packets sent by other nodes or the time of arrival (TOA) of pulses emitted by their neighbors [1]. Both of these proxies allow for distance estimation between pairs of nodes; thus, we can attempt to reconstruct the network topology. This is often termed self-localization [2]- [4]. The molecular conformation problem is another instance of a distance problem [5], and so is reconstructing a room's geometry from echoes [6]. Less obviously, sparse phase retrieval [7] can be converted to a distance problem and addressed using EDMs.Sometimes the data are not metric, but we seek a metric representation, as it happens commonly in psychometrics [8]. As a matter of fact, the psychometrics community is at the root of the development of a number of tools related to EDMs, including multidimensional scaling (MDS)-the problem of finding the best point set representation of a given set of distances. More abstractly, we can study EDMs for objects such as images, which live in highdimensional vector spaces [9].EDMs are a useful description of the point sets and a starting point for algorithm design. A typical task is to retrieve the original point configuration: it may initially come as a surprise that this requires no more than an eigenvalue decomposition (EVD) of a symmetric matrix. In fact, the majority of Euclidean distance problems requi...

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The embedding problem for predistance matrices

Cited by 32 publications

References 25 publications

DINDSCAL: direct INDSCAL

DINDSCAL: direct INDSCAL

Positive semidefinite relaxations for distance geometry problems

Euclidean Distance Matrices: Essential theory, algorithms, and applications

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