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## Eigenvector synchronization, graph rigidity and the molecule problem (2012)

Citations: | 14 - 9 self |

### Citations

3721 | Normalized cuts and image segmentation - Shi, Malik |

965 | Closed-form solution of absolute orientation using unit quaternions - Horn - 1987 |

571 |
Least-squares fitting of two 3-D point sets
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(Show Context)
Citation Context ...ormation. For that reason, we estimate hi by the closest orthogonal matrix to h̃i in the Frobenius matrix norm 6 ĥi = argmin X∈O(3) ‖h̃i −X‖F (3.9) We do this by using the well known procedure (e.g.,=-=[3]-=-), ĥi = UiV T i , where h̃i = UiΣiV T i is the singular value decomposition of h̃i, see also [22] and [38]. Note that the estimation of the orthogonal transformations of the patches are up to a globa... |

555 | A tutorial on spectral clustering - Luxburg - 2007 |

387 | Robust distributed network localization with noisy range measurements - Moore, Leonard, et al. - 2004 |

248 | Graph implementations for nonsmooth convex programs
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Citation Context ... we also consider Algorithm 2 in our simulations only for computational reasons, since the running time of the FULL-SDP algorithm is significantly smaller compared to our CVX-based SDP implementation =-=[28, 27]-=- of problem (4.3). Our intuition about the usefulness of the WUL subgraphs is supported by several numerical simulations. Figure 4.1 and Table 4.1 show the reconstruction errors of the patches (in ter... |

177 | Improved MDS-Based Localization - Shang, Ruml, et al. - 2004 |

173 | Embeddability of weighted graphs in k-space is strongly NP-hard - Saxe - 1979 |

153 | Conditions for unique graph realizations - HENDRICKSON - 1992 |

145 | Sensor positioning in wireless ad-hoc sensor networks using multidimensional scaling”, inIEEE INFOCOM - Ji, Zha - 2004 |

132 | Sensor networks: an overview - Tubaishat, Madria - 2003 |

119 | A theory of network localization
- Aspnes, Eren, et al.
(Show Context)
Citation Context ... received a great deal of attention in the networking and distributed computation communities, and numerous heuristic algorithms exist that approximate its solution. In the context of sensor networks =-=[36, 5, 4, 2]-=-, there are many algorithms that solve the graph realization problem, and they include methods such as global optimization [14], semidefinite programming (SDP) [13, 9, 10, 50, 51, 61] and local to glo... |

110 | Semidefinite programming based algorithms for sensor network localization
- BISWAS, LIAN, et al.
(Show Context)
Citation Context ...realization problem has attracted a lot of attention in recent years, across many communities. The problem and its variants come up naturally in a variety of settings such as wireless sensor networks =-=[9, 53]-=-, structural biology [30], dimensionality reduction, Euclidean ball packing and multidimensional scaling (MDS) [19]. In such real world applications, the given distances dij between adjacent nodes are... |

100 | Connected rigidity matroids and unique realizations of graphs - Jackson, Jordán |

92 | Semidefinite programming approaches for sensor network localization with noisy distance measurements
- Biswas, Liang, et al.
- 2006
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Citation Context ...context of sensor networks [36, 5, 4, 2], there are many algorithms that solve the graph realization problem, and they include methods such as global optimization [14], semidefinite programming (SDP) =-=[13, 9, 10, 50, 51, 61]-=- and local to global approaches [42, 45, 40, 47, 60]. A popular model for the graph realization problem is that of a geometric graph model, where the distance between two nodes is available if and onl... |

88 | On the computational complexity of sensor network localization
- Aspnes, Goldberg, et al.
- 2004
(Show Context)
Citation Context ... received a great deal of attention in the networking and distributed computation communities, and numerous heuristic algorithms exist that approximate its solution. In the context of sensor networks =-=[36, 5, 4, 2]-=-, there are many algorithms that solve the graph realization problem, and they include methods such as global optimization [14], semidefinite programming (SDP) [13, 9, 10, 50, 51, 61] and local to glo... |

76 |
Validation of protein structure from anisotropic carbonyl chemical shifts in a dilute liquid crystalline phase
- CORNILESCU, MARQUARDT, et al.
- 1998
(Show Context)
Citation Context ...51 8.95 40% 0.01292 0.01120 19.65 14.5 7.62 13.42 13.55 12.8 50% 0.01639 0.01462 24.57 18.1 9.82 16.93 18.24 17.35 Table 6.2 Denoising distances at various levels of noise for the ubiqutin (PDG 1d3z, =-=[18]-=-). η̄SDP,GOOD denotes the average noise of the hard constraint (“good”) distances after the SDP embeddings. Note that the current implementation of the SNL-SDP algorithm used for the patch embeddings ... |

73 | Applications of convex analysis to multidimensional scaling - Leeuw - 1977 |

72 | Generic global rigidity
- Connelly
- 2005
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Citation Context ...nodes pseudo-anchors. Two known necessary conditions for global rigidity in R3 are 4-connectivity and redundant rigidity (meaning the graph remains locally rigid after the removal of any single edge) =-=[29, 16]-=-. One approach to breaking up a patch graph into globally rigid subgraphs, used by the ABBIE algorithm of [30], is to recurse on each 4-connected component of the graph, and then on each redundantly r... |

67 | The molecule problem: exploiting structure in global optimization - Hendrickson - 1995 |

59 |
Some metric inequalities in the space of matrices
- Fan, Hoffmann
- 1955
(Show Context)
Citation Context ...s matrix norm 6 ĥi = argmin X∈O(3) ‖h̃i −X‖F (3.9) We do this by using the well known procedure (e.g.,[3]), ĥi = UiV T i , where h̃i = UiΣiV T i is the singular value decomposition of h̃i, see also =-=[22]-=- and [38]. Note that the estimation of the orthogonal transformations of the patches are up to a global orthogonal transformation (i.e., a global rotation and reflection with respect to the original e... |

50 | Distributed clock synchronization over wireless networks: Algorithm and analysis
- Giridhar, Kumar
- 2006
(Show Context)
Citation Context ...ction is large enough, we obtain measurements for the ratios of the unknown group elements. Finding group elements from noisy measurements of their ratios is also known as the synchronization problem =-=[37, 24]-=-. For example, the synchronization of clocks in a distributed network from noisy measurements of their time offsets is a particular example of synchronization over R. [48] introduced an eigenvector me... |

45 | Groenen, Modern Multidimensional Scaling: Theory and Applications - Borg, F - 2005 |

44 | Graph laplacian regularization for large-scale semidefinite programming - WEINBERGER, SHA, et al. |

42 | Angular synchronization by eigenvectors and semidefinite programming - Singer |

41 | A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization
- BISWAS, YE
- 2003
(Show Context)
Citation Context ...onal challenges posed by the limitations of the SDP solvers, several divide and conquer approaches have been proposed recently for the graph realization problem. One of the earlier methods appears in =-=[12]-=-, and more recent methods include the Distributed Anchor Free Graph Localization (DAFGL) algorithm of [11], and the DISCO algorithm (DIStributed COnformation) of [41]. One of the critical assumptions ... |

39 | Characterizing generic global rigidity
- GORTLER, HEALY, et al.
(Show Context)
Citation Context ... its neighbors together with the node itself, and by G(i) = (V (i), E(i)) its subgraph of 1-hop neighbors. IfG(i) is globally rigid, which can be checked efficiently using the randomized algorithm of =-=[25]-=-, then we embed it in R3. Using SDP for globally rigid (sub)graphs can still produce incorrect localizations, even for noiseless data. In order to ensure that SDP would give the correct localization, ... |

39 | Optimal and global time synchronization in sensornets - Karp, Elson, et al. - 2003 |

35 | Distributed Graph Layout for Sensor Networks
- Gotsman, Koren
- 2004
(Show Context)
Citation Context ...ify globally rigid subgraphs of its 1-hop neighborhood that we call patches. Each patch is then separately localized in a coordinate system of its own using either the stress minimization approach of =-=[26]-=-, or by using semidefinite programming (SDP). 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, th... |

32 | ARIA2: automated NOE assignment and data integration in NMR structure calculation. Bioinformatics - Rieping, Habeck, et al. |

32 | Three-dimensional structure determination from common lines in cryo-EM by eigenvectors and semidefinite programming, submitted for publication - Singer, Shkolnisky |

32 | Between min cut and graph bisection - Wagner, Wagner - 1993 |

31 | The HADDOCK web server for data-driven biomolecular docking, Nature Protocols 5 - VRIES, DIJK, et al. - 2010 |

30 | A remark on global positioning from local distances - Singer - 2008 |

23 | A Cheeger inequality for the graph connection laplacian,” available online - Bandeira, Singer, et al. - 2012 |

23 | Sensor network localization by eigenvector synchronization over the Euclidean group
- CUCURINGU, LIPMAN, et al.
- 2011
(Show Context)
Citation Context ...this paper, we focus on the molecular distance geometry problem (to which we will refer from now on as the molecule problem) in R3, although the approach is applicable to other dimensions as well. In =-=[20]-=- we introduced 2D-ASAP, an eigenvector based synchronization algorithm that solves the sensor network localization problem in R2. We summarize below the approach used in 2D-ASAP, and further explain t... |

22 | Leeuw, “Applications of convex analysis to multidimensional scaling,” in Recent Developments in Statistics - de - 1977 |

21 | NMR studies of structure and function of biological macromolecules - Wuthrich |

19 | Localization from incomplete noisy distance measurements, submitted - JAVANMARD, MONTANARI - 2011 |

18 | DILAND: An algorithm for distributed sensor localization with noisy distance measurements - KHAN, KAR, et al. |

17 |
Weak alignment NMR: a hawk-eyed view of biomolecular structure
- BAX, GRISHAEV
(Show Context)
Citation Context ...s. One other important information which we are ignoring is given by the residual dipolar couplings (RDC) measurements that give noisy angle information (cos2(θ)) with respect to a global orientation =-=[7]-=-. Another approach is to consider an energy based formulation that captures the interaction between atoms in a readily computable fashion, such as the Lennard-Jones potential. One may then use this in... |

16 | Global rigidity: The effect of coning
- CONNELLY, WHITELEY
(Show Context)
Citation Context ... dimensional combinatorial global rigidity have been established only recently, and motivated our approach for building patches in 2D-ASAP [34, 29]. Due to the recent coning result in rigidity theory =-=[17]-=-, it is also possible to extract globally rigid patches in dimension three. However, such globally rigid patches cannot always be localized accurately by SDP algorithms, even in the case of noiseless ... |

16 | An As-Rigid-As-Possible approach to sensor network localization - ZHANG, LIU, et al. - 2011 |

14 | Closest unitary, orthogonal and hermitian operators to a given operator - KELLER - 1975 |

14 |
Extending the power of quantum chemistry to large systems with the fragment molecular orbital method
- FEDOROV, KITAURA
(Show Context)
Citation Context ... we are trying to localize. For example, in the NMR application, there are often subsets of atoms (referred to as “molecular fragments”, by analogy with the fragment molecular orbital approach, e.g., =-=[23]-=-) whose relative coordinates are known a priori, and thus it is desirable to be able to incorporate such information in the reconstruction process. Of course, one may always input into the problem all... |

12 |
Semidefinite programming algorithms for sensor network localization using angle of arrival information
- BISWAS, AGHAJAN, et al.
- 2005
(Show Context)
Citation Context ...s and numerical algorithms have been proposed in the last decade. A popular approach to solving the graph realization problem is based on SDP, and has attracted considerable attention in recent years =-=[13, 8, 9, 10, 61]-=-. We defer to Section 4.4 a description of existing SDP relaxations of the graph realization problem. Such SDP techniques are usually able to localize accurately small to medium sized problems (up to ... |

12 | A distributed SDP approach for large scale noisy anchor-free graph realization with applications to molecular conformation - BISWAS, TOH, et al. |

12 | An SDP-based divide-and-conquer algorithm for large scale noisy anchor-free graph realization - Leung, Toh |

12 | Universal rigidity: Towards accurate and efficient localization of wireless networks - ZHU, SO, et al. - 2010 |

9 | PATCHWORK: Efficient localization for sensor networks by distributed global optimization, tech - KOREN, GOTSMAN, et al. - 2005 |

7 | Some theoretical aspects of location-location problems - YEMINI - 1979 |

5 | Some unusual matrix eigenvalue problems
- BAI, GOLUB
- 1999
(Show Context)
Citation Context ...S − S)z − 2Ua+ 2λz = 0 which can be written as (DS − S + λI)z = Ua (7.9) Using the fact that zT z = l, (7.9) becomes (Ua)T (DS − S + λI) −2(Ua) = l (7.10) which we solve for λ. We refer the reader to =-=[6]-=- for a more detailed analysis of eigenvalue solutions to such quadratically constrained optimization problems. Finally, the solution to the minimization problem formulated in (7.7) is given by z∗ = (D... |

4 |
Graphical properties of easily localizable networks, Wireless Networks
- ANDERSON, BELHUMEUR, et al.
(Show Context)
Citation Context ... received a great deal of attention in the networking and distributed computation communities, and numerous heuristic algorithms exist that approximate its solution. In the context of sensor networks =-=[36, 5, 4, 2]-=-, there are many algorithms that solve the graph realization problem, and they include methods such as global optimization [14], semidefinite programming (SDP) [13, 9, 10, 50, 51, 61] and local to glo... |

4 | On efficient semidefinite relaxations for quadratically constrained quadratic programmig, Thesis for the degree of - Ding - 2007 |

3 | SNLSDP version 0 - a MATLAB software for sensor network localization - TOH, BISWAS, et al. |

2 | grPartition a MATLAB function for graph partitioning - HESPANHA |

1 | proteomics project - Ab, Atkinson, et al. - 2006 |