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...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...

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...|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...

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... 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...

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...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...

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...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...

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...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...

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....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...

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...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...

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...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 ...

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...,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 ...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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... −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...

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...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...

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...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...