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## Localization from Incomplete Noisy Distance Measurements,” (2011)

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Venue: | in IEEE ISIT |

Citations: | 20 - 0 self |

### Citations

2461 |
A global geometric framework for nonlinear dimensionality reduction.
- Tenenbaum, Silva, et al.
- 2000
(Show Context)
Citation Context ...its variants have attracted significant interest over the past years due to their applications in numerous areas, such as sensor network localization [5], NMR spectroscopy [13], and manifold learning =-=[18, 22]-=-, to name a few. Of particular interest to our work are the algorithms proposed for the localization problem [15, 20, 5, 23]. In general, few performance guarantees have been proved for these algorith... |

1226 | Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation
- Belkin, Niyogi
- 2003
(Show Context)
Citation Context ...|E| ≤ 3/4n 2 Kdr d . Using the bounds on σmax(Ω), σmin(Ω) and |E| in Theorem 5.1 yields the thesis. 125.1 Proof of Theorem 5.2 Before turning to the proof, it is worth mentioning that the authors in =-=[3]-=- propose a heuristic argument showing Ωv ≈ L 2 v for smoothly varying vectors v. Since σmin(L) ≥ C(nr d )r 2 (see Remark 2.2), this heuristic supports the claim of the theorem. In the following, we fi... |

545 |
Random Geometric Graphs
- Penrose
- 2003
(Show Context)
Citation Context ... the localization problem to be solvable. It is a well known result that the graph G(n,r) is connected w.h.p. if Kdr d > (log n+cn)/n, where Kd is the volume of the d-dimensional unit ball and cn → ∞ =-=[17]-=-. Vice versa, the graph is disconnected with positive probability if Kdr d ≤ (log n + C)/n for some constant C. Hence, we focus on the regime where r ≥ α(log n/n) 1/d for some constant α. We further n... |

385 | Think globally, fit locally: Unsupervised learning of low dimensional manifolds.
- Saul, Roweis, et al.
- 2003
(Show Context)
Citation Context ...its variants have attracted significant interest over the past years due to their applications in numerous areas, such as sensor network localization [5], NMR spectroscopy [13], and manifold learning =-=[18, 22]-=-, to name a few. Of particular interest to our work are the algorithms proposed for the localization problem [15, 20, 5, 23]. In general, few performance guarantees have been proved for these algorith... |

305 | Locating the nodes: cooperative localization in wireless sensor networks
- Patwari, Ash, et al.
- 2005
(Show Context)
Citation Context ...nsive distributed measurements. Energy and hardware constraints rule out the use of global positioning systems, and several proposed systems exploit pairwise distance measurements between the sensors =-=[16, 14]-=-. These techniques have acquired new industrial interest due to their relevance to indoor positioning. In this context, global positioning systems are not a method of choice because of their limited a... |

275 |
Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data
- Donoho, Grimes
- 2003
(Show Context)
Citation Context ...P methods to manifold learning. It is typically assumed that the manifold Md is isometrically equivalent to a region in Rd . For the sake of simplicity we shall assume that this region is convex (see =-=[11]-=- for a discussion of this point). With little loss of generality we can indeed identify the region with the unit hypercube [−0.5,0.5] d . A typical manifold learning algorithm ([22] and [23]) estimate... |

224 | Semidefinite Programming for Ad Hoc Wireless Sensor Network Localization,”
- Biswas, Ye
- 2004
(Show Context)
Citation Context ....1. 1.3 Related work The localization problem and its variants have attracted significant interest over the past years due to their applications in numerous areas, such as sensor network localization =-=[5]-=-, NMR spectroscopy [13], and manifold learning [18, 22], to name a few. Of particular interest to our work are the algorithms proposed for the localization problem [15, 20, 5, 23]. In general, few per... |

209 | Wireless sensor network localization techniques.
- Mao, Fidan, et al.
- 2007
(Show Context)
Citation Context ...nsive distributed measurements. Energy and hardware constraints rule out the use of global positioning systems, and several proposed systems exploit pairwise distance measurements between the sensors =-=[16, 14]-=-. These techniques have acquired new industrial interest due to their relevance to indoor positioning. In this context, global positioning systems are not a method of choice because of their limited a... |

204 | Comparison theorems for reversible markov chains.
- Diaconis, Saloff-Coste
- 1993
(Show Context)
Citation Context ...tions which will be used in the proof. Claim 5.2. There exists a constant C = C(d), such that, w.h.p., L ≼ C n∑ k=1 P ⊥ u . Ck The argument is closely related to the Markov chain comparison technique =-=[10]-=-. The proof is given in Appendix D. The next claim provides a concentration result about the number of nodes in the cliques Ci. Its proof is immediate and deferred to Appendix E. Claim 5.3. For every ... |

125 | Graph approximations to geodesics on embedded manifolds.
- Bernstein, Silva, et al.
- 2001
(Show Context)
Citation Context ...,t {‖¨γ(t)‖}, r0 where γ varies over all unit-speed geodesics in M and t is in the domain of γ. For instance, an Euclidean sphere of radius r0 has minimum radius of curvature equal to r0. As shown in =-=[4]-=- (Lemma 3), (1 − d2 ij /24r2 0 )dij ≤ ˜ dij ≤ dij. Therefore, |zij| ∝ d4 ij /r2 0 , and ∆ ∝ r4 /r2 0 . Theorem 1.1 supports the claim that the estimation error d(X, ˆ X) is bounded by C(nrd ) 5 /r2 0 ... |

123 | A theory of network localization,”
- Aspnes, Eren, et al.
- 2006
(Show Context)
Citation Context ...then w.h.p., the SDP-based algorithm recovers the exact positions (up to rigid transformations). In particular, the random geometric graph G(n, r) is w.h.p. globally rigid if r ≥ 10√d(log n/n)1/d. In =-=[3]-=-, the authors prove a similar result on global rigidity of G(n, r). Namely, they show that if n points are drawn from a Poisson process in [0, 1]2, then the random geometric graph G(n, r) is globally ... |

120 | Theory of semidefinite programming for sensor network localization,”
- So, Ye
- 2007
(Show Context)
Citation Context ...thms in the second group formulate the localization problem as a non-convex optimization problem and then use different relaxation schemes to solve it. An example of this type is relaxation to an SDP =-=[5, 21, 24, 1, 23]-=-. A crucial assumption in these works is the existence of some anchors among the nodes whose exact positions are known. The SDP is then used to efficiently check whether the graph is uniquely d-locali... |

105 |
The rigidity of graphs.
- Asimow, Roth
- 1978
(Show Context)
Citation Context ...es of the points up to rigid transformations. This section is a very brief overview of definitions and results in rigidity theory which will be useful in this paper. We refer the interested reader to =-=[12, 2]-=-, for a thorough discussion. A framework GX in R d is an undirected graph G = (V,E) along with a configuration X ∈ R n×d which assigns a point xi ∈ R d to each vertex i of the graph. The edges of G co... |

82 | Solving Euclidean distance matrix completion problems via semidefinite programming.
- Alfakih, Khandani, et al.
- 1999
(Show Context)
Citation Context ...thms in the second group formulate the localization problem as a non-convex optimization problem and then use different relaxation schemes to solve it. An example of this type is relaxation to an SDP =-=[5, 21, 24, 1, 23]-=-. A crucial assumption in these works is the existence of some anchors among the nodes whose exact positions are known. The SDP is then used to efficiently check whether the graph is uniquely d-locali... |

73 | Generic global rigidity.
- Connelly
- 2005
(Show Context)
Citation Context ...the points do not satisfy any nonzero polynomial equation with integer coefficients). The connection between global rigidity and stress matrices is demonstrated in the following two results proved in =-=[8]-=- and [12]. Theorem 2.1 (Connelly, 2005). If X is a generic configuration in R d with a stress matrix Ω of rank n − d − 1, then GX is globally rigid in R d . Theorem 2.2 (Gortler, Healy, Thurston, 2010... |

40 | Characterizing generic global rigidity,”
- Gortler, Healy, et al.
- 2010
(Show Context)
Citation Context ...es of the points up to rigid transformations. This section is a very brief overview of definitions and results in rigidity theory which will be useful in this paper. We refer the interested reader to =-=[12, 2]-=-, for a thorough discussion. A framework GX in R d is an undirected graph G = (V,E) along with a configuration X ∈ R n×d which assigns a point xi ∈ R d to each vertex i of the graph. The edges of G co... |

40 | An introduction to nonlinear dimensionality reduction by maximum variance unfolding
- Weinberger, Saul
- 2006
(Show Context)
Citation Context ...h as sensor network localization [5], NMR spectroscopy [13], and manifold learning [18, 22], to name a few. Of particular interest to our work are the algorithms proposed for the localization problem =-=[15, 20, 5, 23]-=-. In general, few performance guarantees have been proved for these algorithms, in particular in the presence of noise. The existing algorithms can be categorized in to two groups. The first group con... |

38 | Mixing TImes for Random Walks on Geometric Random Graphs
- Boyd, Ghosh, et al.
- 2005
(Show Context)
Citation Context ...tric graph G(n,r), defined as Ln = D −1/2 LD −1/2 , where D is the diagonal matrix with degrees of the nodes on diagonal. Then, w.h.p., λ2(Ln), the second smallest eigenvalue of Ln, is at least Cr 2 (=-=[6, 17]-=-). Also, using the result of [7] (Theorem 4) and Corollary 2.1, we have λ2(L) ≥ C(nr d )r 2 , for some constant C = C(d). 2.3 Notations For a vector v ∈ R n , and a subset T ⊆ {1, · · · ,n}, vT ∈ R T ... |

32 | Framework for kernel regularization with application to protein clustering. In
- Fan, Keles, et al.
- 2005
(Show Context)
Citation Context ...he localization problem and its variants have attracted significant interest over the past years due to their applications in numerous areas, such as sensor network localization [5], NMR spectroscopy =-=[13]-=-, and manifold learning [18, 22], to name a few. Of particular interest to our work are the algorithms proposed for the localization problem [15, 20, 5, 23]. In general, few performance guarantees hav... |

30 | A remark on global positioning from local distances.
- Singer
- 2008
(Show Context)
Citation Context ...h as sensor network localization [5], NMR spectroscopy [13], and manifold learning [18, 22], to name a few. Of particular interest to our work are the algorithms proposed for the localization problem =-=[15, 20, 5, 23]-=-. In general, few performance guarantees have been proved for these algorithms, in particular in the presence of noise. The existing algorithms can be categorized in to two groups. The first group con... |

20 | Sensor network localization from local connectivity: Performance analysis for the mds-map algorithm.
- Oh, Karbasi, et al.
- 2010
(Show Context)
Citation Context ...tion matrices X and Y are called equivalent up to rigid transformation, if there exists O ∈ O(d) and a shift s ∈ Rd such that Y = XO + usT . We use the following metric, similar to the one defined in =-=[16]-=-, to evaluate the distance between the original position matrix X ∈ Rn×d and the estimation X̂ ∈ Rn×d. Let L = I−(1/n)uuT be the centering matrix . Note that L is an n×n symmetric matrix of rank n−1 w... |

14 | Universal rigidity: Towards accurate and efficient localization of wireless networks,”
- Zhu, So, et al.
- 2010
(Show Context)
Citation Context ...thms in the second group formulate the localization problem as a non-convex optimization problem and then use different relaxation schemes to solve it. An example of this type is relaxation to an SDP =-=[5, 21, 24, 1, 23]-=-. A crucial assumption in these works is the existence of some anchors among the nodes whose exact positions are known. The SDP is then used to efficiently check whether the graph is uniquely d-locali... |

9 |
Eigenvalues and structures of graphs.
- Butler
- 2008
(Show Context)
Citation Context ... −1/2 LD −1/2 , where D is the diagonal matrix with degrees of the nodes on diagonal. Then, w.h.p., λ2(Ln), the second smallest eigenvalue of Ln, is at least Cr 2 ([6, 17]). Also, using the result of =-=[7]-=- (Theorem 4) and Corollary 2.1, we have λ2(L) ≥ C(nr d )r 2 , for some constant C = C(d). 2.3 Notations For a vector v ∈ R n , and a subset T ⊆ {1, · · · ,n}, vT ∈ R T is the restriction of v to indic... |

9 |
Multidimensional Scaling. Monographs on Statistics and Applied Probability 88. Chapman and Hall,
- Cox, Cox
- 2001
(Show Context)
Citation Context ...be categorized in to two groups. The first group consists of algorithms who try first to estimate the missing distances and then use MDS to find the positions from 3the reconstructed distance matrix =-=[15, 9]-=-. MDS-MAP [9] and ISOMAP [22] are two well-known examples of this class where the missing entries of the distance matrix are approximated by computing the shortest paths between all pairs of nodes. Th... |

2 |
On sensor network localization using SDP relaxation. arXiv:1010.2262
- Shamsi, Ye, et al.
- 2010
(Show Context)
Citation Context ...dity of G(n,r). As a special case of Theorem 1.1 we can consider the problem of reconstructing the point positions from exact measurements. The case of exact measurements was also studied recently in =-=[19]-=- following a different approach. This corresponds to setting ∆ = 0. The underlying question is whether the point positions {xi}i∈V can be efficiently determined (up to a rigid motion) by the set of di... |