In this section, we show how to use the approach of this section to solve the spectral relax- ation of the permutation synchronization problem in a decentralized fashion.
Recall from Chapter 3, that permutation synchronization can be cast as the following com- binatorial optimization problem:
maximize Π1,...,Πn∈Sm
X
{i,j}∈E
tr(ΠTi ΠeijΠj) (5.41)
intractability of problem (3.5), we, next, propose a spectral relaxation. In contrast to the spectral relaxation of [101], we do not assume that all pairwise association matricesΠeij are
available.
The following lemma is crucial for deriving the spectral relaxation of problem (3.5).
Lemma 5.6.1. Let ∆= diag(. Π1e ). Assume that the pairwise associations {Πeij}{i,j}∈E are
consistent and the sensor graph G is connected. Then, the matrix ∆−1 e
Π has exactly m
leading eigenvalues equal to 1, wherem is the size of the universe of features. Furthermore, there exist permutation matrices Π1,Π2, . . . ,Πn ∈ Pm, unique up to a global permutation,
satisfying
e
ΠΠ = ∆Π, (5.42)
where Π = [ΠT1 ΠT2 · · · ΠTn]T.
For a proof of Lemma 5.6.1 we refer the reader to [7].
An approximate solution to problem (5.41), under relaxed orthonormality and nonnegativity constraints, is determined by the m leading eigenvectors of ∆−1Πe [7]. Since, ∆−1Πe is not, in general, a symmetric matrix, the approach of this chapter is not directly applicable. Nevertheless, ∆−1Πe is similar to the symmetric matrix ∆−1/2Πe∆−1/2. Let VΛVT be an eigendecomposition of ∆−1/2Πe∆−1/2, where V is an orthogonal matrix and Λ a diagonal. Then, let ∆−1/2V = Q1 .. . Qn . =Q, (5.43)
for some m×m matrices Q1, Q2, . . . , Qn. In the noiseless case, span(Π) and span(Q) are
equal, and thus, Π = QG−1 for some invertible matrix G. Without loss of generality, we can assume thatΠ1 =Im and thus,Gmust be equal to Q1. Based on this observation, the last step of the proposed approach consists of each agent i computing the approximately
optimal Π?i by
Π?i = ProjPm(QiQ −1
1 ), (5.44)
whereProjPm denotes the projection onto set of m×mpermutation matrices which can be
computed using the Hungarian algorithm [67]. Finally, cycle consistent pairwise associations can be computed by
Π?ij = Π?iΠ?Tj . (5.45)
Overall, we propose the following four-step approach for solving the spectral relaxation of permutation synchronization in a decentralized fashion:
1. All agents collectively compute themleading eigenvectorsV = [v1 · · · vm]of∆−1/2Πe∆−1/2 using the approach of Section 5.4.
2. Each agenticomputes Qi as defined in (5.43).
3. Agent 1transmits Q1 to the entire group. This operation takes diam(G)time steps.
4. Each agenticomputesΠ?i as in (5.44) and pairs of agents compute their corresponding associations by (5.45).
Finally, we evaluate the propose spectral relaxation. We consider three graph topologies, namely, a six-regular, a ten-regular graph and a complete graph, all of which have n = 20 vertices. We used m = 30 as the number of features per collection. We vary the percentage of outliers of the initial pairwise associations{eπij}{i,j}∈Efrom 0% to 90%. Results
are presented in Fig. 5.4. We observe that, not surprisingly, increasing the connectivity of a graph, significantly improves the accuracy of the spectral relaxation for permutation synchronization. Furthermore, given enough noisy pairwise associations, exact recovery of the true pairwise associations is achieved for a significant percentage of outliers, namely about 60%.
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Percentage of outliers
Asso
ci
ation
accuracy
6-Regular 10-Regular CompleteFigure 5.4: Accuracy of pairwise associations after permutation synchronization versus initial percentage of outliers for a six-regular, a ten-regular and a complete graph with 20 vertices. We observe that in the six-regular graph case, exact recovery of the true pairwise associations can be achieved for up to 20% of outliers, whereas this percentage increases to 40% for the case of the 10-regular graph and to 60% for the case of a complete graph.
5.7
Conclusions
In this chapter, we proposed a dynamical systems approach to distributedly compute any number of extreme eigenvalues and associated eigenvectors of a matrix that is distributed across a network with almost-global convergence guarantees. In contrast to approaches based on Orthogonal Iteration, orthogonality constraints are only asymptotically satisfied by the dynamical system herein proposed. Thus, the main computational burden of the Decentralized Orthogonal Iteration, namely, the orthonormalization step, is not present in our approach. In addition, we applied the proposed method to permutation synchronization, specifically to decentralize the spectral relaxation of the permutation synchronization.
Chapter 6
Distributed cooperative state
estimation for mobile agents
6.1
Introduction
In this chapter, we first propose a method for fusion of two random vectors with unknown cross-correlations which is less conservative than the widely used Covariance Intersection (CI) while taking cross-correlations into account. Then, we extend our formulation for the case of a linear measurement model. Finally, we present numerical examples and simulations, in a distributed cooperative localization scenario, which demonstrate the validity of the proposed approach and that the proposed approach significantly outperforms Covariance Intersection, while taking correlations into account.
This chapter is structured as follows: in Section 6.2.1 we include definitions of consistency and related notions and we introduce the problem at hand. Our game-theoretic approach to fusing two random variables with unknown cross-correlations is the topic of Section 6.2.2 and it is generalized for arbitrary linear measurement models in Section 6.3. In Section 6.4 we include details on the implemented numerical algorithm. Numerical examples and
simulation results are presented in Sections 6.5 and 6.6 respectively.