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ASAP: An Eigenvector Synchronization Algorithm for the Graph Realization Problem

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Distance Geometry

Abstract

We review a recent algorithm for localization of points in Euclidean space from a sparse and noisy subset of their pairwise distances. Our approach starts by extracting and embedding uniquely realizable subsets of neighboring sensors called patches. In the noise-free case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation, and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. In other words, to every patch, there corresponds an element of the Euclidean group Euc(3) of rigid transformations in \({\mathbb{R}}^{3}\), and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The algorithm is scalable as the number of nodes increases, and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity and running time.

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Notes

  1. 1.

    We used the SNL-SDP code of [44].

  2. 2.

    For example three common vertices, although the precise definition of “enough” will be given later.

  3. 3.

    The diagonal matrix D should not be confused with the partial distance matrix.

  4. 4.

    Not to be confused with G(i) = (V (i), E(i)) defined in the beginning of this section.

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Authors and Affiliations

  1. PACM, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Mihai Cucuringu

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  1. Mihai Cucuringu
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Correspondence to Mihai Cucuringu .

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Editors and Affiliations

  1. IRISA, University of Rennes 1, avenue du General Leclerc, Rennes, 35042, France

    Antonio Mucherino

  2. , Dept of Applied Maths (IMECC-UNICAMP), State University of Campinas, Campinas, 13081, Brazil

    Carlile Lavor

  3. , LIX, Ecole Polytechnique, Palaiseau, 91128, France

    Leo Liberti

  4. Instituto Alberto Luiz Coimbra de, Pos-Graduacao e Pesquisa de, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, Brazil

    Nelson Maculan

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Cucuringu, M. (2013). ASAP: An Eigenvector Synchronization Algorithm for the Graph Realization Problem. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds) Distance Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5128-0_10

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