674 research outputs found
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
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
, 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 , 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
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
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
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
The angular synchronization problem aims to accurately estimate (up to a
constant additive phase) a set of unknown angles from 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 -synchronization) is to estimate 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
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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