# Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

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## Abstract

We study Semidefinite Programming, * SDP*, relaxations for Sensor Network Localization,

*, with anchors and with noisy distance information. The main point of the paper is to view*

**SNL***as a (nearest) Euclidean Distance Matrix,*

**SNL***, completion problem that does not distinguish between the anchors and the sensors. We show that there are advantages for using the well studied*

**EDM***model. In fact, the set of anchors simply corresponds to a given fixed clique for the graph of the*

**EDM***problem.*

**EDM**We next propose a method of projection when large cliques or dense subgraphs are identified. This projection reduces the size, and improves the stability, of the relaxation. In addition, by viewing the problem as an * EDM* completion problem, we are able to derive a new approximation scheme for the sensors from the

*approximation. This yields, on average, better low rank approximations for the low dimensional realizations. This further emphasizes the theme that*

**SDP***is in fact just an*

**SNL***problem.*

**EDM**We solve the * SDP* relaxations using a primal-dual interior/exterior-point algorithm based on the Gauss-Newton search direction. By not restricting iterations to the interior, we usually get lower rank optimal solutions and thus, better approximations for the

*problem. We discuss the relative stability and strength of two formulations and the corresponding algorithms that are used. In particular, we show that the quadratic formulation arising from the*

**SNL***relaxation is better conditioned than the linearized form that is used in the literature.*

**SDP**## Keywords

Sensor Network Localization Anchors Graph realization Euclidean Distance Matrix completions Semidefinite Programming## Preview

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