Eigenvector synchronization, graph rigidity and the molecule problem

Mihai Cucuringu, Amit Singer, David Cowburn

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in R3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in R3, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

Original languageEnglish (US)
Pages (from-to)21-67
Number of pages47
JournalInformation and Inference
Volume1
Issue number1
DOIs
StatePublished - Jan 1 2012

Fingerprint

Eigenvalues and eigenfunctions
Rigidity
Eigenvector
Synchronization
Molecules
Patch
Graph in graph theory
Partitioning
Biology
Subgraph
Distance measurement
Distance Measurement
Divide and conquer
Wireless sensor networks
Substructure
Positioning
Preprocessing
Wireless Sensor Networks
Euclidean
Connectivity

Keywords

  • Conquer
  • Distance geometry
  • Divide
  • Eigenvectors
  • Graph realization
  • Rigidity theory
  • SDP
  • Spectral graph theory
  • Synchronization
  • The molecule problem

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Analysis
  • Applied Mathematics
  • Statistics and Probability
  • Numerical Analysis

Cite this

Eigenvector synchronization, graph rigidity and the molecule problem. / Cucuringu, Mihai; Singer, Amit; Cowburn, David.

In: Information and Inference, Vol. 1, No. 1, 01.01.2012, p. 21-67.

Research output: Contribution to journalArticle

Cucuringu, Mihai ; Singer, Amit ; Cowburn, David. / Eigenvector synchronization, graph rigidity and the molecule problem. In: Information and Inference. 2012 ; Vol. 1, No. 1. pp. 21-67.
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