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## Sensor network localization by eigenvector synchronization over the Euclidean group

Venue: | In press |

Citations: | 23 - 14 self |

### Citations

2381 | Nonlinear dimensionality reduction by locally linear embedding
- Roweis, Saul
- 2000
(Show Context)
Citation Context ...scale SDP by a much smaller SDP which leads to a significant reduction in running time. The Locally Rigid Embedding (LRE) algorithm [Singer 2008] is reminiscent of the Locally Linear Embedding (LLE) [=-=Roweis and Saul 2000-=-] technique used in machine learning for dimensionality reduction. LRE tries to preserve, in a global coordinatesystem, thelocalaffinerelationshipspresentwithin patches. Eachsensor contributes with a ... |

1193 | Laplacian eigenmaps for dimensionality reduction and data representation - Belkin, Niyogi - 2003 |

965 | Closed-form solution of absolute orientation using unit quaternions
- Horn
- 1987
(Show Context)
Citation Context ...overlap is often small. It is therefore crucial to have robust alignment methods even when the overlap size is small. A closed form solution to the registration problem in any dimension was given by [=-=Horn et al. 1988-=-], where the best rigid transformation between two sets of points is obtained by various matrix manipulations and eigenvalue/eigenvector decomposition. In our approach described in the following parag... |

695 |
Modern multidimensional scaling: Theory and applications
- Borg, Groenen
- 1997
(Show Context)
Citation Context ...s observed to perform well in practice compared to other traditional optimization methods is a variant of the gradient descent approach called the stress majorization algorithm, also known as SMACOF [=-=Borg and Groenen 2005-=-], originally introduced by De Leeuw [1977]. The main drawback of this approach is that the stress function in (3) is not convex and the search for the global minimum is prone to getting stuck at loca... |

432 | Numerical linear algebra - Trefethen, Bau - 1997 |

387 | Robust distributed network localization with noisy range measurements
- Moore, Leonard, et al.
- 2004
(Show Context)
Citation Context ... global optimization [Borg and Groenen 2005], semidefinite programming (SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; So and Ye 2005; Zhu et al. 2010] and local to global approaches [=-=Moore et al. 2004-=-; Shang and Ruml 2004; Koren et al. 2005; Singer 2008; Zhang et al. 2010]. In thispaper, wefocus onthe problemofsensornetworklocalization(SNL) inthe plane (R 2 ), although the approach is applicable t... |

245 |
Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State
- Frank
(Show Context)
Citation Context ...ization problem of sensor networks. In particular, in [Singer and Shkolnisky ; Hadani and Singer ] we demonstrated its usefulness in solving the “class averaging” problem in cryo-electron microscopy [=-=Frank 2006-=-], and showed its mathematical connection to the parallel transport and the connection-Laplacian operators from differential geometry. We are currently extending this approach to the 3D structure from... |

223 | Semidefinite programming for ad hoc wireless sensor network localization - Biswas, Ye - 2004 |

177 | Improved MDS-Based Localization
- Shang, Ruml, et al.
- 2004
(Show Context)
Citation Context ...n [Borg and Groenen 2005], semidefinite programming (SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; So and Ye 2005; Zhu et al. 2010] and local to global approaches [Moore et al. 2004; =-=Shang and Ruml 2004-=-; Koren et al. 2005; Singer 2008; Zhang et al. 2010]. In thispaper, wefocus onthe problemofsensornetworklocalization(SNL) inthe plane (R 2 ), although the approach is applicable to higher dimensions d... |

173 |
Embeddability of weighted graphs in k-space is strongly NP-hard
- Saxe
- 1979
(Show Context)
Citation Context ...izable (see, e.g., [Hendrickson 1992]). The graph realization problem is known to be difficult; Saxe has shown it is strongly NP-complete in one dimension, and strongly NP-hard for higher dimensions [=-=Saxe 1979-=-; Yemini 1979]. Despite its difficulty, the graph realization problem has received a great deal of attention in the networking and distributed computation communities, and numerous heuristic algorithm... |

153 | Conditions for unique graph realizations
- HENDRICKSON
- 1992
(Show Context)
Citation Context ...an embedding exists, it is unique (up to rigid transformations) only if there are enough distance constraints, in which case the graph is said to be globally rigid or uniquely realizable (see, e.g., [=-=Hendrickson 1992-=-]). The graph realization problem is known to be difficult; Saxe has shown it is strongly NP-complete in one dimension, and strongly NP-hard for higher dimensions [Saxe 1979; Yemini 1979]. Despite its... |

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
(Show Context)
Citation Context ... ding p1,p2,...,pn ∈ R d such that ‖pi − pj‖ = dij, for all (i,j) ∈ E. The graph realization problem comes up naturally in a variety of settings such as wireless sensor networks [Biswas et al. 2006a; =-=Tubaishat and Madria 2003-=-], structural biology [Hendrickson 1995] and multidimensional scaling (MDS) [Cox and Cox 2001]. In such real world applications, the given distances dij between nodes are not accurate, dij = ‖pi −pj‖+... |

129 | Image deformation using moving least squares
- Schaefer, McPhail, et al.
(Show Context)
Citation Context ...od of least squares and find rθ and t. By solving the registration problem using complex least squares, we are guaranteed to recover the optimal solution (best 2D rigid transformation) up to scaling [=-=Schaefer et al. 2006-=-]. For noisy distance measurements, rθ does not necessarily lie on the unit circle, in which case we extract its phase θ (but ignore its amplitude). For noisy data, the registration method becomes sig... |

119 | A theory of network localization
- Aspnes, Eren, et al.
(Show Context)
Citation Context ... the networking and distributed computation communities, and numerous heuristic algorithms exist that approximate its solution. In the context of sensor networks [Ji and Zha 2004; Aspnes et al. 2004; =-=Aspnes et al. 2006-=-; Anderson et al. 2009], there are many algorithms that solve the graph realization problem, and they include methods such as global optimization [Borg and Groenen 2005], semidefinite programming (SDP... |

118 | Theory of Semidefinite Programming for Sensor Network Localization
- So, Yu
- 2005
(Show Context)
Citation Context ...e the graph realization problem, and they include methods such as global optimization [Borg and Groenen 2005], semidefinite programming (SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; =-=So and Ye 2005-=-; Zhu et al. 2010] and local to global approaches [Moore et al. 2004; Shang and Ruml 2004; Koren et al. 2005; Singer 2008; Zhang et al. 2010]. In thispaper, wefocus onthe problemofsensornetworklocaliz... |

110 | Semidefinite programming based algorithms for sensor network localization
- BISWAS, LIAN, et al.
(Show Context)
Citation Context ...ihai Cucuringu et al. ding p1,p2,...,pn ∈ R d such that ‖pi − pj‖ = dij, for all (i,j) ∈ E. The graph realization problem comes up naturally in a variety of settings such as wireless sensor networks [=-=Biswas et al. 2006-=-a; Tubaishat and Madria 2003], structural biology [Hendrickson 1995] and multidimensional scaling (MDS) [Cox and Cox 2001]. In such real world applications, the given distances dij between nodes are n... |

92 | Semidefinite programming approaches for sensor network localization with noisy distance measurements
- Biswas, Liang, et al.
- 2006
(Show Context)
Citation Context ...ihai Cucuringu et al. ding p1,p2,...,pn ∈ R d such that ‖pi − pj‖ = dij, for all (i,j) ∈ E. The graph realization problem comes up naturally in a variety of settings such as wireless sensor networks [=-=Biswas et al. 2006-=-a; Tubaishat and Madria 2003], structural biology [Hendrickson 1995] and multidimensional scaling (MDS) [Cox and Cox 2001]. In such real world applications, the given distances dij between nodes are n... |

88 | On the computational complexity of sensor network localization - Aspnes, Goldberg, et al. - 2004 |

83 | Localization and routing in sensor networks by local angle information
- BRUCK, GAO, et al.
- 2005
(Show Context)
Citation Context ... to noise is another possible generalization. The inclusion of angular measurements in localization algorithms has not been studied as thoroughly as distance measurements, but recent papers, such as [=-=Bruck et al. 2009-=-], provide insight for this potential improvement. In the context of the ASAP algorithm, angular measurements should lead to better localization and alignment of the patches. There are a few possible ... |

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

67 | The molecule problem: exploiting structure in global optimization
- Hendrickson
- 1995
(Show Context)
Citation Context ...ij, for all (i,j) ∈ E. The graph realization problem comes up naturally in a variety of settings such as wireless sensor networks [Biswas et al. 2006a; Tubaishat and Madria 2003], structural biology [=-=Hendrickson 1995-=-] and multidimensional scaling (MDS) [Cox and Cox 2001]. In such real world applications, the given distances dij between nodes are not accurate, dij = ‖pi −pj‖+εij where εij represents the added nois... |

62 | The fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem
- SUN, BOYD, et al.
(Show Context)
Citation Context ...iated in Section 4 and by the intuition that faster mixing of the random walk should assist the eigenvector method to integrate and propagate consistency relations over cycles in the patch graph. In [=-=Sun et al. 2006-=-] it is shown that this fastest mixing design problem is convex and can be solved using SDP, which interestingly enough draws similarities with the MVU approach to localization. The synchronization pr... |

50 | Distributed clock synchronization over wireless networks: Algorithm and analysis - Giridhar, Kumar - 2006 |

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

42 | Angular synchronization by eigenvectors and semidefinite programming
- Singer
(Show Context)
Citation Context ... We explain in more detail in Section 6 the procedures by which we align pairs of patches, if such an alignment is at all possible. Prior to computing the top eigenvector of the matrix Z, as done in [=-=Singer 2010-=-], we choose to normalize it as follows. Let D be an N ×N diagonal matrix, 4 whose entries are given by Dii = ∑ N j=1 |zij|. In other words, Dii = deg(i), (5) where deg(i) is the node degree of patch ... |

39 | Optimal and global time synchronization in sensornets
- Karp, Elson, et al.
- 2003
(Show Context)
Citation Context ... of linear equations. 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 [=-=Karp et al. 2003-=-; Giridhar and Kumar 2006]. 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. Singer ... |

35 | Distributed Graph Layout for Sensor Networks
- Gotsman, Koren
- 2004
(Show Context)
Citation Context ...mber of nodes in G, since the neighborhood graph of a node may contribute several patches or none. For the embedding of local patches we usually use the Stress majorization algorithm as described in [=-=Gotsman and Koren 2004-=-]. Once each patch is embedded in its own coordinate system, one must find the reflections, rotations and translations that will stitch all patches together in a consistent manner, a process to which ... |

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

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

27 | Viewing angle classification of cryo-electron microscopy images using eigenvectors, submitted for publication - Singer, Shkolnisky, et al. |

16 | Global rigidity: The effect of coning - CONNELLY, WHITELEY |

16 | An As-Rigid-As-Possible approach to sensor network localization
- ZHANG, LIU, et al.
- 2011
(Show Context)
Citation Context ...SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; So and Ye 2005; Zhu et al. 2010] and local to global approaches [Moore et al. 2004; Shang and Ruml 2004; Koren et al. 2005; Singer 2008; =-=Zhang et al. 2010-=-]. In thispaper, wefocus onthe problemofsensornetworklocalization(SNL) inthe plane (R 2 ), although the approach is applicable to higher dimensions d > 2 as well. Sensor networks are a collection of a... |

15 | Orientability and diffusion map - Singer, Wu |

12 |
Semidefinite programming algorithms for sensor network localization using angle of arrival information
- BISWAS, AGHAJAN, et al.
- 2005
(Show Context)
Citation Context ... all measured distances dij,(i,j) ∈ E as best as possible. A popular approach to solving the SNL problem is based on SDP, and has attracted considerable attention in recent years [Biswas and Ye 2004; =-=Biswas et al. 2005-=-; Biswas et al. 2006a; 2006b; Zhu et al. 2010]. One possible way of solving the SNL problem is to find the embedding p1,...,pn that minimizes the following error function ∑ ( min ‖pi −pj‖ 2 −d 2 ) 2. ... |

12 | A semidefinite programming approach to the graph realization problem: Theory, applications and extensions
- SO
- 2007
(Show Context)
Citation Context ...that solve the graph realization problem, and they include methods such as global optimization [Borg and Groenen 2005], semidefinite programming (SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; =-=So 2007-=-; So and Ye 2005; Zhu et al. 2010] and local to global approaches [Moore et al. 2004; Shang and Ruml 2004; Koren et al. 2005; Singer 2008; Zhang et al. 2010]. In thispaper, wefocus onthe problemofsens... |

12 | Universal rigidity: Towards accurate and efficient localization of wireless networks
- ZHU, SO, et al.
- 2010
(Show Context)
Citation Context ...ization problem, and they include methods such as global optimization [Borg and Groenen 2005], semidefinite programming (SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; So and Ye 2005; =-=Zhu et al. 2010-=-] and local to global approaches [Moore et al. 2004; Shang and Ruml 2004; Koren et al. 2005; Singer 2008; Zhang et al. 2010]. In thispaper, wefocus onthe problemofsensornetworklocalization(SNL) inthe ... |

9 | PATCHWORK: Efficient localization for sensor networks by distributed global optimization, tech
- KOREN, GOTSMAN, et al.
- 2005
(Show Context)
Citation Context ...005], semidefinite programming (SDP) [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; So and Ye 2005; Zhu et al. 2010] and local to global approaches [Moore et al. 2004; Shang and Ruml 2004; =-=Koren et al. 2005-=-; Singer 2008; Zhang et al. 2010]. In thispaper, wefocus onthe problemofsensornetworklocalization(SNL) inthe plane (R 2 ), although the approach is applicable to higher dimensions d > 2 as well. Senso... |

7 |
Some theoretical aspects of location-location problems
- YEMINI
- 1979
(Show Context)
Citation Context ..., e.g., [Hendrickson 1992]). The graph realization problem is known to be difficult; Saxe has shown it is strongly NP-complete in one dimension, and strongly NP-hard for higher dimensions [Saxe 1979; =-=Yemini 1979-=-]. Despite its difficulty, the graph realization problem has received a great deal of attention in the networking and distributed computation communities, and numerous heuristic algorithms exist that ... |

5 | A linear-space algorithm for distance preserving graph embedding
- ASANO, BOSE, et al.
(Show Context)
Citation Context ...t most k −1 patches contributed by the 1-hop neighborhood of any node. The running time for localizing the patches depends on the embedding method of choice, SMACOF or FULSDP. In terms ofcomplexity, [=-=Asano et al. 2009-=-]show that the SMACOF algorithm runs in O(k 3 +k 3/2 t) time and O(k) space, where t is the number of iterations required to minimize the stress energy function introduced earlier in (3). In our exper... |

4 |
Graphical properties of easily localizable networks, Wireless Networks
- ANDERSON, BELHUMEUR, et al.
(Show Context)
Citation Context ...distributed computation communities, and numerous heuristic algorithms exist that approximate its solution. In the context of sensor networks [Ji and Zha 2004; Aspnes et al. 2004; Aspnes et al. 2006; =-=Anderson et al. 2009-=-], there are many algorithms that solve the graph realization problem, and they include methods such as global optimization [Borg and Groenen 2005], semidefinite programming (SDP) [Biswas and Ye 2004;... |

3 |
SNLSDP version 0 - a MATLAB software for sensor network localization
- TOH, BISWAS, et al.
(Show Context)
Citation Context ...8 , it produces similar results in terms of the localization error, but with lower running times (see Figure 15). To the best of our knowledge, the SDP based approaches 8 We used the SNL-SDP code of [=-=Toh et al. 2008-=-]. ACM Journal Name, Vol. x, No. y, x 2010.24 · Mihai Cucuringu et al. (in particular those of [Biswas and Ye 2004; Biswas et al. 2006a; 2006b; So 2007; So and Ye 2005; Zhu et al. 2010] ) have not be... |

3 | for Embedded Networked Sensing - rep - 2003 |

1 | Closed-formsolution ofabsolute orientation using orthonormal matrices - Horn, Hilden, et al. |

1 | y, x 2010. network localization by synchronization · 41 - No - 2004 |

1 |
User’s guide for sedumi interface 1.04. www.laas.fr/~peaucell/software/sdmguide.ps.gz
- Peaucelle, Henrion, et al.
- 2002
(Show Context)
Citation Context ...y of SeDuMi (the SDP solver used here) is O(k 2 m ′2.5 + m ′3.5 ), since there are k number of decision variables (nodes of a patch) and m ′ linear matrix (in)equality constraints (edges of a patch) [=-=Peaucelle et al. 2002-=-]. In either scenario, the embedding of a single patch remains polynomial in k, and the complexity of the preprocessing step adds up to O(N poly(k)), since there are N patches for which we check bicon... |

1 | Image deformationusingmoving leastsquares - Schaefer, Mcphail, et al. |

1 | Article 19, Publication date: July 2012. Network Localization by Eigenvector Synchronization - Anal |

1 | y, x 2010. Sensor network localization by synchronization · 41 - No - 2004 |