### A new graph parameter related to bounded rank positive semidefinite matrix completions

- MATHEMATICAL PROGRAMMING

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### Quantum strategic game theory

- In Proceedings of the 3rd Innovations in Theoretical Computer Science
, 2012

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### Comparing SOS and SDP relaxations of sensor network localization

, 2010

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### 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.

### RELAX AND UNFOLD: MICROPHONE LOCALIZATION WITH EUCLIDEAN DISTANCE MATRICES

"... Recent methods for microphone position calibration work with sound sources at a priori unknown locations. This is convenient for ad hoc arrays, as it requires little additional infrastructure. We propose a flexible localization algorithm by first recognizing the problem as an instance of multidimens ..."

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Recent methods for microphone position calibration work with sound sources at a priori unknown locations. This is convenient for ad hoc arrays, as it requires little additional infrastructure. We propose a flexible localization algorithm by first recognizing the problem as an instance of multidimensional unfolding (MDU)—a classical problem in Euclidean geometry and psychometrics—and then solving the MDU as a special case of Euclidean distance matrix (EDM) completion. We solve the EDM completion using a semidef-inite relaxation. In contrast to existing methods, the semidefinite formulation allows us to elegantly handle missing pairwise distance information, but also to incorporate various prior information about the distances between the pairs of microphones or sources, bounds on these distances, or ordinal information such as “microphones 1 and 2 are more apart than microphones 1 and 15”. The intuition that this should improve the localization performance is justified by numerical experiments. Index Terms—Microphone localization, array calibration, mi-crophone array, Euclidean distance matrix, semidefinite relaxation, multidimensional unfolding 1.

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Note that the pdf version of this report contains links to related information.

### 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.

### RELAX AND UNFOLD: MICROPHONE LOCALIZATIONWITH EUCLIDEAN DISTANCE MATRICES

"... Recent methods for localization of microphones in a microphone array exploit sound sources at a priori unknown locations. This is convenient for ad-hoc arrays, as it requires little additional in-frastructure. We propose a flexible localization algorithm by first recognizing the problem as an instan ..."

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Recent methods for localization of microphones in a microphone array exploit sound sources at a priori unknown locations. This is convenient for ad-hoc arrays, as it requires little additional in-frastructure. We propose a flexible localization algorithm by first recognizing the problem as an instance of multidimensional un-folding (MDU)—a classical problem in Euclidean geometry and psychometrics—and then solving the MDU as a special case of Euclidean distance matrix (EDM) completion. We solve the EDM completion using a semidefinite relaxation. In contrast to existing methods, the semidefinite formulation allows us to elegantly han-dle missing pairwise distance information, but also to incorporate various prior information about the distances between the pairs of microphones or sources, bounds on these distances, or ordinal information such as “microphones 1 and 2 are more apart than mi-crophones 1 and 15”. The intuition that this should improve the localization performance is confirmed by numerical experiments. Index Terms—Microphone localization, array calibration, mi-crophone array, Euclidean distance matrix, semidefinite relaxation, multidimensional unfolding 1.