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**1 - 6**of**6**### A Lagrangian Dual Approach to the Single-Source Localization Problem

, 2013

"... The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approa ..."

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The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approach. It is proved that the Lagrangian dual problem must have an optimal solution under the generalized Slater condition. We provide a sufficient condition for the zero-duality gap and establish the equivalence between the Lagrangian dual approach and the existing Generalized Trust-Region Subproblem (GTRS) approach studied by Beck et al. [3]. We also reveal new implications of the assumptions made by the GTRS approach. Moreover, the Lagrangian dual approach has a straightforward extension to the multiple-source localization problem. Numerical simulations demonstrate that the Lagrangian dual approach can produce localization of similar quality as the GTRS and can significantly outperform the well-known semidefinite programming solver SNLSDP for the multiple source localization problem on the tested cases.

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

"... 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 learn-ing, wireless sensor networks, acoustics, and more. Despite ..."

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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 learn-ing, 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.

### GreMuTRRR: A Novel Genetic Algorithm to Solve Distance Geometry Problem for Protein Structures

"... Nuclear Magnetic Resonance (NMR) Spectroscopy is a widely used technique to predict the native struc-ture of proteins. However, NMR machines are only able to report approximate and partial distances between pair of atoms. To build the protein structure one has to solve the Euclidean distance geometr ..."

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Nuclear Magnetic Resonance (NMR) Spectroscopy is a widely used technique to predict the native struc-ture of proteins. However, NMR machines are only able to report approximate and partial distances between pair of atoms. To build the protein structure one has to solve the Euclidean distance geometry problem given the incomplete interval distance data produced by NMR machines. In this paper, we propose a new genetic algorithm for solving the Euclidean distance geometry problem for protein structure prediction given sparse NMR data. Our genetic algorithm uses a greedy mutation operator to intensify the search, a twin removal technique for diversification in the population and a random restart method to recover stagnation. On a standard set of benchmark dataset, our algorithm significantly outperforms standard genetic algorithms.

### Fisher information distance: a geometrical reading∗

, 2014

"... This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissimilarity between two probability distribution functions. The Fisher distance, as well as other divergence measures, are also used in many applications to establish a proper data average. The main purpose ..."

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This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissimilarity between two probability distribution functions. The Fisher distance, as well as other divergence measures, are also used in many applications to establish a proper data average. The main purpose is to widen the range of possible interpretations and relations of the Fisher distance and its associated geometry for the prospective applications. It focuses on statistical models of the normal probability distribution functions and takes advantage of the connection with the classical hyperbolic geometry to derive closed forms for the Fisher distance in several cases. Connections with the well-known Kullback-Leibler divergence mea-sure are also devised.

### Computational experience on Distance Geometry Problems 2.0

, 2014

"... We propose a set of formulations and reformulations of the Distance Geometry Problem, which we evaluate with both local and global off-the-shelf solvers. The local solvers are cast in a global optimization metaheuristic (Variable Neighbourhood Search) since the problem is nonconvex and non-global op ..."

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We propose a set of formulations and reformulations of the Distance Geometry Problem, which we evaluate with both local and global off-the-shelf solvers. The local solvers are cast in a global optimization metaheuristic (Variable Neighbourhood Search) since the problem is nonconvex and non-global optima are usually of limited practical interest. 1