We consider the problem of localizing wireless devices in an ad-hoc network embedded in a d-dimensional Euclidean space. Obtaining a good estimation of where wireless devices are located is crucial in wireless network applications including environment monitoring, geographic routing and topology control. When the positions of the devices are unknown and only local distance information is given, we need to infer the positions from these local distance measurements. This problem is particularly challenging when we only have access to measurements that have limited accuracy and are incomplete. We consider the extreme case of this limitation on the available information, namely only the connectivity information is available, i.e., we only know whether a pair of nodes is within a fixed detection range of each other or not, and no information is known about how far apart they are. Further, to account for detection failures, we assume that even if a pair of devices is within the detection range, it fails to detect the presence of one another with some probability and this probability of failure depends on how far apart those devices are. Given this limited information, we investigate the performance of a centralized positioning algorithm MDS-MAP introduced by , and a distributed positioning algorithm, introduced by , called HOP-TERRAIN. In particular, for a network consisting of n devices positioned randomly, we provide a bound on the resulting error for both algorithms. We show that the error is bounded, decreasing at a rate that is proportional to R Rc, where Rc is the critical detection range when the resulting random network starts to be connected, and R is the detection range of each device.
Of particular interest are the two new results on the performance of sensor localization algorithms. In @cite_41 , proposes a new reconstruction algorithm based on semidefinite programming where they could establish lower and upper bounds on the reconstruction errors of their algorithm. Similarly, in @cite_9 , due to new advances in matrix completion methods @cite_28 , the authors analyse the performance of OptSpace @cite_36 , a novel matrix completion algorithm, in localizing the sensors. Interestingly, they did not need to adhere to the assumptions made by @cite_17 . However, they have a restrictive assumption about the topology of the network, specifically, sensors are scattered inside an annulus.
@cite_41 @cite_28 @cite_9 @cite_36 @cite_17
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This paper focuses on the principled study of distance reconstruction for distance-based node localization. We address an important issue in node localization by showing that a highly incomplete set of inter-node distance measurements obtained in ad-hoc node deployments carries sufficient information for the accurate reconstruction of the missing distances, even in the presence of noise and sensor node failures. We provide an efficient and provably accurate algorithm for this reconstruction, and we show that the resulting error is bounded, decreasing at a rate that is inversely proportional to radicn, the square root of the number of nodes in the region of deployment. Although this result is applicable to many localization schemes, in this paper we illustrate its use in conjunction with the popular multidimensional scaling algorithm. Our analysis reveals valuable insights and key factors to consider during the sensor network setup phase, to improve the quality of the position estimates The sensor network localization (SNL) problem in embedding dimension @math consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming (SDP) completion problem by using the linear mapping between Euclidean distance matrices (EDM) and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem. In this paper, we improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. (1) In compressed sensing, we show that if the m×n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tractable l 1 minimization even if a positive fraction of the measurements are arbitrarily corrupted, provided the number of nonzero entries in x is O(m (log(n m)+1)). (2) In the very general sensing model introduced in Candes and Plan (IEEE Trans. Inf. Theory 57(11):7235–7254, 2011) and assuming a positive fraction of corrupted measurements, exact recovery still holds if the signal now has O(m (log2 n)) nonzero entries. (3) Finally, we prove that one can recover an n×n low-rank matrix from m corrupted sampled entries by tractable optimization provided the rank is on the order of O(m (nlog2 n)); again, this holds when there is a positive fraction of corrupted samples. We present two algorithms for large-scale noisy low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and partly parallelizable. Our second algorithm is a first-order method for maximizing the trace---a popular low-rank inducing regularizer---in the formulation of the problem with a constrained misfit. Both of the methods output a point configuration that can serve as a high-quality initialization for local optimization techniques. Numerical experiments on large-scale sensor localization problems illustrate the two approaches.