674 research outputs found

    Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

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    The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in R3\mathbb{R}^3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. 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, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in R3\mathbb{R}^3, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

    Cram\'er-Rao bounds for synchronization of rotations

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    Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise

    IoT Localization and Optimized Topology Extraction Using Eigenvector Synchronization

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    Internet-of-Things (IoT) devices are low size, weight and power (SWaP), low complexity and include sensors, meters, wearables and trackers. Transmitting information with high signal power is exacting on device battery life, therefore an efficient link and network configuration is absolutely crucial to avoid signal power enhancement in interference-rich environment and resorting to battery-life extending strategies. Efficient network configuration can also ensure fulfilment of network performance metrics like throughput, coding rate and spectral efficiency. We formulate a novel approach of first localizing the IoT nodes and then extracting the network topology for information exchange between the nodes (devices, gateway and sinks), such that overall network throughput is maximized. The nodes are localized using noisy measurements of a subset of Euclidean distances between two nodes. Realizable subsets of neighboring devices agree with their own position within the entire network graph through eigenvector synchronization. Using communication global graph-model-based technique, network topology is constructed in terms of transmit power allocation with the aim of maximizing spatial usage and overall network throughput. This topology extraction problem is solved using the concept of linear programming

    Concentration of the Kirchhoff index for Erdos-Renyi graphs

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    Given an undirected graph, the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors. Summing these distances over all pairs of nodes yields the so-called Kirchhoff index of the graph, which measures its overall connectivity. In this work, we consider Erdos-Renyi random graphs. Since the graphs are random, their Kirchhoff indices are random variables. We give formulas for the expected value of the Kirchhoff index and show it concentrates around its expectation. We achieve this by studying the trace of the pseudoinverse of the Laplacian of Erdos-Renyi graphs. For synchronization (a class of estimation problems on graphs) our results imply that acquiring pairwise measurements uniformly at random is a good strategy, even if only a vanishing proportion of the measurements can be acquired

    Robust Angular Synchronization via Directed Graph Neural Networks

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    The angular synchronization problem aims to accurately estimate (up to a constant additive phase) a set of unknown angles θ1,…,θn∈[0,2π)\theta_1, \dots, \theta_n\in[0, 2\pi) from mm noisy measurements of their offsets \theta_i-\theta_j \;\mbox{mod} \; 2\pi. Applications include, for example, sensor network localization, phase retrieval, and distributed clock synchronization. An extension of the problem to the heterogeneous setting (dubbed kk-synchronization) is to estimate kk groups of angles simultaneously, given noisy observations (with unknown group assignment) from each group. Existing methods for angular synchronization usually perform poorly in high-noise regimes, which are common in applications. In this paper, we leverage neural networks for the angular synchronization problem, and its heterogeneous extension, by proposing GNNSync, a theoretically-grounded end-to-end trainable framework using directed graph neural networks. In addition, new loss functions are devised to encode synchronization objectives. Experimental results on extensive data sets demonstrate that GNNSync attains competitive, and often superior, performance against a comprehensive set of baselines for the angular synchronization problem and its extension, validating the robustness of GNNSync even at high noise levels

    Collaborative Perception From Data Association To Localization

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    During the last decade, visual sensors have become ubiquitous. One or more cameras can be found in devices ranging from smartphones to unmanned aerial vehicles and autonomous cars. During the same time, we have witnessed the emergence of large scale networks ranging from sensor networks to robotic swarms. Assume multiple visual sensors perceive the same scene from different viewpoints. In order to achieve consistent perception, the problem of correspondences between ob- served features must be first solved. Then, it is often necessary to perform distributed localization, i.e. to estimate the pose of each agent with respect to a global reference frame. Having everything set in the same coordinate system and everything having the same meaning for all agents, operation of the agents and interpretation of the jointly observed scene become possible. The questions we address in this thesis are the following: first, can a group of visual sensors agree on what they see, in a decentralized fashion? This is the problem of collaborative data association. Then, based on what they see, can the visual sensors agree on where they are, in a decentralized fashion as well? This is the problem of cooperative localization. The contributions of this work are five-fold. We are the first to address the problem of consistent multiway matching in a decentralized setting. Secondly, we propose an efficient decentralized dynamical systems approach for computing any number of smallest eigenvalues and the associated eigenvectors of a weighted graph with global convergence guarantees with direct applications in group synchronization problems, e.g. permutations or rotations synchronization. Thirdly, we propose a state-of-the art framework for decentralized collaborative localization for mobile agents under the presence of unknown cross-correlations by solving a minimax optimization prob- lem to account for the missing information. Fourthly, we are the first to present an approach to the 3-D rotation localization of a camera sensor network from relative bearing measurements. Lastly, we focus on the case of a group of three visual sensors. We propose a novel Riemannian geometric representation of the trifocal tensor which relates projections of points and lines in three overlapping views. The aforemen- tioned representation enables the use of the state-of-the-art optimization methods on Riemannian manifolds and the use of robust averaging techniques for estimating the trifocal tensor
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