Distributed methods for synchronization of orthogonal matrices over graphs

TL;DR

This paper introduces two novel distributed algorithms for synchronizing orthogonal matrices over directed graphs, with guarantees of convergence and stability under various graph conditions, applicable to 3D localization and image registration.

Contribution

The paper presents two new algorithms for orthogonal matrix synchronization over directed graphs, one for symmetric and one for asymmetric graphs, with proven convergence and stability properties.

Findings

Algorithms converge under general conditions.

Methods are stable for small step sizes.

Numerical simulations verify effectiveness.

Abstract

This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as a least-squares optimization problem with nonlinear constraints. The synchronization problem appears as one of the key components in applications ranging from 3D-localization to image registration. The main contributions of this work can be summarized as the introduction of two novel algorithms; one for symmetric graphs and one for graphs that are possibly asymmetric. Under general conditions, the former has guaranteed convergence to the solution of a spectral relaxation to the synchronization problem. The latter is stable for small step sizes when the graph is quasi-strongly connected. The proposed methods are verified in numerical simulations.

Tables

Table 1. Table 1: Conditions for convergence.

(1)	$𝒢=(𝒱,ℰ)$ is connected and symmetric.
(2)	$R_{ij}\in O\left(d\right)$ for all $(i,j)\in ℰ$.
(3)	${\left\{R_{ij}\right\}}_{(i,j)\in ℰ}$ is transitively consistent.
(4)	, where
	$P=\text{diag}\left(\left(A+A^T\right)1_n\right)+A+A^T$.
(5)	, where
	$\overline{L}=\text{diag}\left(\left(A+A^T\right)1_n\right)-\left(A+A^T\right)$.
(6)	${\overline{R}}_i\in GL(d,ℝ)$ for all the ${\overline{R}}_i$-matrices in $\overline{X}$.
(7)	${\sum }_{i\in 𝒱}{\overline{R}}_i\in GL(d,ℝ),$ where the ${\overline{R}}_i$ are the
	matrices in $\overline{X}$.
(8)	It holds that ${\lambda }_{\left(n-1\right)d}>{\lambda }_{\left(n-1\right)d+1},$
	where ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$ are the eigenvalues
	of $L_{\text{undir}}$, i.e., $L_{\text{undir}}=V\text{diag}\left([{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{nd}]\right)V^T$.
(9)	It holds that ${\lambda }_{nd-\left(i+1\right)}>{\lambda }_{nd-i},$ for
	$i=0,1,\dots d-2$ where the ${\lambda }_i$ are
	defined in (8) above.
(10)	, where $L$ is the graph Laplacian
	matrix of the graph $𝒢$ defined in Section 2
	(not to mix up with $\overline{L}$).

Table 2. Table 2: Conditions for convergence.

(1)	$𝒢=(𝒱,ℰ)$ is QSC.
(2)	$𝒢=(𝒱,ℰ)$ is strongly connected.
(3)	$R_{ij}\in O\left(d\right)$ for all $(i,j)\in ℰ$.
(4)	${\left\{R_{ij}\right\}}_{(i,j)\in ℰ}$ is transitively consistent.
(5)	Let $\text{Re}\left({\lambda }_1\right)\le \text{Re}\left({\lambda }_2\right)\le \cdots \le \text{Re}\left({\lambda }_n\right)$,
	where the ${\lambda }_i$’s are the eigenvalues
	of $\overline{L}=\text{diag}\left(A{1}_n\right)-A$. It holds that
	for $i=1,2,\dots ,n-1$.
(6)	There are ${\delta }_1,{\delta }_2>0$ such that ${\delta }_1\le a_{ij}\le {\delta }_2$
	for all the $a_{ij}$-weights.

Equations

$${\mathcal{N}}_i=\{j:(i,j)\in \mathcal{E}\}.$$

$$A_{ij}=\{\begin{array}{ll}1& \text{ if }(i,j)\in \mathcal{E},\\ 0& \text{ else.}\end{array}$$

$$L=\text{diag}(A{1}_n)-A,$$

$$((i,j)\in \mathcal{E})\Rightarrow ((j,i)\in \mathcal{E})\text{ for all }(i,j)\in \mathcal{V}\times \mathcal{V}.$$

$$A(\bar{\mathcal{G}})=(A(\mathcal{G}){)}^T\text{ and }L(\bar{\mathcal{G}})=\text{diag}((A(\mathcal{G}){)}^T{1}_n)-A(\mathcal{G}{)}^T.$$

$$O(d)=\{R\in {\mathbb{R}}^{d\times d}:R^{T}R=I_d\}.$$

$$R_{ik}=R_{ij}{R}_{jk}$$

$$R_{ij}=R_i^{-1}{R}_j$$

$$Q{R}_i={\bar{R}}_i\text{ for all }i.$$

$$R_{ij}=R_i^{-1}{R}_j\text{ for all }(i,j)\in \mathcal{E},$$

$$(P_1)\{\begin{array}{l}\begin{array}{rlrl}& \underset{R}{\mathrm{minimize}}& & f_1(R)=\sum _{(i,j)\in \mathcal{E}}\frac{a_{ij}}{2}\Vert R_{ij}-R_i^T{R}_j{\Vert }_F^2,\\ & \text{subject to}& & R=[R_1,R_2,\dots ,R_n],\\ & & & R_i\in O(d)\text{ for all }i,\end{array}\end{array}$$

$$f_1(U_1)$$

$$U_1$$

$$L_{\text{undir}}=\text{diag}(A{1}_n)\otimes I_d+\text{diag}({\bar{W}}^T\bar{W})-(W+W^T),$$

$${\tilde{R}}_i(k)$$

$$+{ϵ}_1\sum _{j\in {\mathcal{N}}_i}(Q_{ij}{\tilde{R}}_j(k-1)-V_{ij}{\tilde{R}}_i(k-1)),$$

$$R_i^T(k)$$

$${\tilde{d}}_{is}(k)$$

$$d_{is}(k)$$

$$+{ϵ}_2\sum _{l\in {\mathcal{N}}_i}(d_{ls}(k-1)-d_{is}(k-1))$$

$$\text{ for }s=1,2,\dots ,d,$$

$$D_i(k)$$

$${\tilde{Q}}_i(k)$$

$$Q_i^T(k)$$

$${\tilde{R}}_i(k-1{)}^{-1}{\tilde{R}}_i(k)=P_i^{-1}(k){\bar{D}}_i(k)P_i(k).$$

$$\tilde{R}(k+1)$$

$$\text{dim}(\text{ker}(L_{\text{undir}}))\le d.$$

$$\text{im}(U_1)\subset \text{ker}(L_{\text{undir}}).$$

$$U_1^T{L}_{\text{undir}}{U}_1=0,$$

$$\text{dim}(\text{ker}(L_{\text{undir}}))\ge d.$$

$$\bar{X}$$

$$=[(R_k^{-1}{R}_1{)}^{-T},(R_k^{-1}{R}_2{)}^{-T},\dots ,(R_k^{-1}{R}_n{)}^{-T}{]}^T$$

$$\Vert R_{ik}{\bar{Y}}_k-{\bar{Y}}_i{\Vert }_F=0,$$

$$(P_2)\{\begin{array}{l}\begin{array}{rlrl}& \underset{X\in {\mathbb{R}}^{nd\times d}}{\mathrm{minimize}}& & \frac{1}{2}\text{tr}(X^T{L}_{\text{undir}}X)\\ & \text{subject to}& & X^{T}X=n{I}_d.\end{array}\end{array}$$

$${\lambda }_{\max }=\underset{v\in {\mathbb{S}}^{nd}}{\max }{v}^T{L}_{\text{undir}}v,$$

$$(R_i^T(k)R_j(k)\to R_{ij}\text{ as }k\to \infty )\text{ for all }(i,j)\in \mathcal{E},$$

Full text

††thanks: The authors gratefully acknowledge the financial support from the Fonds National de la Recherche, Luxembourg (FNR8864515, FNR6538106).

, ,

1 Introduction

This paper introduces two new distributed algorithms for the problem of synchronizing orthogonal matrices over graphs. Synchronization means that compositions of transformations (multiplications of matrices) over loops in the graph equal the identity (matrix)  [1, 2, 3, 4]. Thus, “synchronization” does not refer to the related concepts of consensus [5] or rendezvous, e.g., attitude synchronization [6]. We formulate the problem as a nonlinear least-squares optimization with matrix variables [7, 8]. For symmetric communication topologies we provide an algorithm with strong convergence guarantees – the solution converges to the optimal solution of a spectral relaxation, which in turn is known to produce near-optimal solutions. For graphs that are possibly asymmetric we provide an algorithm with weaker convergence guarantees but with good performance in numerical simulations.

The synchronization problem appears as one of the key components in the following applications: the 3D-localization problem, where the transformations are obtained from camera measurements; the generalized Procrustes problem, where scales, rotations, and translations are calculated between multiple point clouds [9]; the image registration problem, where transformations are calculated between multiple images [10]. Due to sensor and communication limitations, there is often a need to use distributed protocols for the 3D-localization problem and several approaches have been proposed recently [11, 12, 13]. There are also many other interesting applications for the synchronization problem, see Section 1.2 in [14].

If we exclude the requirement that the synchronization method shall be distributed, there is an extensive body of work. Govindu et al. have presented several approaches based on Lie-group averaging, where a first-order approximation in the tangent space is used [15, 16, 17]. Singer et al. have presented several optimization approaches [1, 2, 3, 18, 19, 20, 21]. Pachauri et al. have addressed the special case where the matrices are permutation matrices [22]. In [3], three types of relaxations of the problem are presented: semidefinite programming relaxation (see [14] for an extensive analysis of this approach); spectral relaxation; least unsquared deviation in combination with semidefinite relaxation. These three relaxations were evaluated in the probabilistic framework where the error to the ground truth was calculated in numerical experiments. The simulations showed that the first two approaches were on par, whereas the last approach performed slightly better. Furthermore, the last approach was significantly more robust to outliers. The first distributed algorithm we present has a connection to the second category of the three relaxations above, since the matrices in the algorithm converge to the optimal solution of the spectral relaxation. Our methods are extrinsic, in the sense that the matrices are calculated in $\mathbb{R}^{d\times d}$ and then projected onto the set of orthogonal matrices. The opposite to extrinsic methods are intrinsic methods where no projections from an ambient space occur. In [23], intrinsic gradient descent methods are studied for the problem of finding the Riemannian center of mass.

The contributions of this work can be summarised as the introduction of two novel algorithms (Algorithm 1 and 2) for distributed synchronization of orthogonal matrices over directed graphs. For both algorithms we provide conditions for guaranteed convergence. The main result of the paper is the above-mentioned convergence in Algorithm 1 to the optimal solution of the spectral relaxation problem (Proposition 14). Previous works in the context of distributed algorithms have focused on undirected graphs and 3D rotations [11, 12, 13]. However, in this work we consider directed graphs and arbitrary dimensions. It should be noted that some of the existing algorithms can be extended to higher dimensions and are given for the 3D-case mostly for clarity of exposition.

The distributed approaches in this work bear a resemblance to linear consensus protocols [24, 25, 26, 27]. The methods also share similarities with the eigenvector method in [28] and gossip algorithms [29]. The important states in our algorithms are matrices, and those combined converge to a tall matrix whose range space is a certain linear subspace. In the case of symmetric communication between agents, the proposed method can either be interpreted as an extension of the power method or the steepest descent method. In our methods, instead of using the graph Laplacian matrix [24], matrices similar to the graph connection Laplacian matrix [30] are used. These matrices can be seen as a generalizations of the graph Laplacian matrix, in which the scalars are replaced by matrix blocks.

The paper proceeds as follows. In Section 2 we introduce the definitions that are necessary in order to precisely state the problem, which is done in Section 3. Subsequently, the distributed method for the case of symmetric graphs (Algorithm 1) is introduced and analyzed in section 4. In Section 5, the distributed method for the case of directed and possibly asymmetric graphs (Algorithm 2) is introduced and analyzed. In Section 6, the paper is concluded.

2 Preliminaries

2.1 Directed Graphs

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a directed graph, where $\mathcal{V}=\{1,2,\ldots,n\}$ is the node set and $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is the edge set. Throughout the paper, the notation $\mathcal{A}\subset\mathcal{B}$ means that every element in $\mathcal{A}$ is contained in $\mathcal{B}$. The set $\mathcal{N}_{i}$ is the set of neighboring nodes of node $i$ and defined by

[TABLE]

The adjacency matrix $A=[A_{ij}]$ for the graph $\mathcal{G}$ is defined by

[TABLE]

The graph Laplacian matrix is defined by

[TABLE]

where $1_{n}\in\mathbb{R}^{n}$ is a vector with all entries equal to $1$. In order to emphasize that the adjacency matrix $A$, the graph Laplacian matrix $L$ and the $\mathcal{N}_{i}$ sets depend on the graph $\mathcal{G}$, we may write $A(\mathcal{G})$, $L(\mathcal{G})$ and $\mathcal{N}_{i}(\mathcal{G})$ respectively. For simplicity however, we mostly omit this notation and simply write $A$, $L$, and $\mathcal{N}_{i}$.

Definition 1**.**

(connected graph, undirected path)

The directed graph $\mathcal{G}$ is connected if there is an undirected path from any node in the graph to any other node. An undirected path is defined as a (finite) sequence of unique nodes such that for any pair $(i,j)$ of consecutive nodes in the sequence it holds that $((i,j)\in\mathcal{E})\text{ or }((j,i)\in\mathcal{E}).$

Definition 2**.**

(quasi-strongly connected graph, center, directed path)

The directed graph $\mathcal{G}$ is quasi-strongly connected (QSC) if it contains a center. A center is a node in the graph to which there is a directed path from any other node in the graph. A directed path is defined as a (finite) sequence of unique nodes such that any pair of consecutive nodes in the sequence comprises an edge in $\mathcal{E}$.

Definition 3**.**

(strongly connected graph)

The directed graph $\mathcal{G}$ is strongly connected if for all pairs of nodes $(i,j)\in\mathcal{V}\times\mathcal{V}$, there is a directed path from $i$ to $j$.

Definition 4**.**

(symmetric graph)

The directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is symmetric if

[TABLE]

Given a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, the graph $\bar{\mathcal{G}}=(\mathcal{V},\bar{\mathcal{E}})$ is the graph constructed by reversing the direction of the edges in $\mathcal{E}$, i.e., $(i,j)\in\bar{\mathcal{E}}$ if and only if $(j,i)\in{\mathcal{E}}$. It is easy to see that

[TABLE]

2.2 Synchronization or transitive consistency of matrices

The set of invertible matrices in $\mathbb{R}^{d\times d}$ is $GL(d,\mathbb{R})$ and the group of orthogonal matrices in $\mathbb{R}^{d\times d}$ is

[TABLE]

The set $SO(d)$ comprises those matrices in $O(d)$ whose determinants are equal to $1$.

Definition 5**.**

(transitive consistency)**

The matrices in the collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{V}\times\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ are transitively consistent for the complete graph if

[TABLE]

*for all $i,j$ and $k$. * 2. 2.

Given a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, the matrices in the collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$ are transitively consistent for $\mathcal{G}$ if there is a collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{V}\times\mathcal{V}}\supset\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ such that $\{{R}_{ij}\}_{(i,j)\in\mathcal{V}\times\mathcal{V}}$ is transitively consistent for the complete graph.

If it is apparent by the context, sometimes we will be less strict and omit to mention which graph a collection of transformations is transitively consistent for. Another word for transitive consistency is synchronization. We will use the two interchangeably. A sufficient condition for synchronization of the $R_{ij}$-matrices for any graph is that there is a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ such that

[TABLE]

for all $(i,j)\in\mathcal{E}$. Lemma 7 below and the proof thereof provides additional important information. The result is similar to that in [12]. For the statement of the lemma, the following definition is needed.

Definition 6**.**

Two collections $\{R_{i}\}_{i\in\mathcal{V}}$ and $\{\bar{R}_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ are equal up to transformation from the left, if there is $Q\in GL(d,\mathbb{R})$ such that

[TABLE]

Lemma 7**.**

For any graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$ that are transitively consistent for $\mathcal{G}$,

there is a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ such that

[TABLE] 2. 2.

all collections $\{R_{i}\}_{i\in\mathcal{V}}$ satisfying (10) are equal up to transformation from the left if and only if $\mathcal{G}$ is connected, 3. 3.

there is a unique collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{V}\times\mathcal{V}}\supset\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of transitively consistent matrices for the complete graph, if and only if all collections $\{R_{i}\}_{i\in\mathcal{V}}$ satisfying (10) are equal up to transformation from the left.

*Proof: * See [31].

Another equivalent definition of transitive consistency or synchronization is given in [12, 32]. A set of transformations is transitively consistent if the composite transformations equal the identity along loops or cycles in the graph. In Proposition 7 in [12] the equivalence between this condition for the loops and (10) is shown. The definition using the auxiliary $R_{i}$-matrices, (10), is the one we will use mostly in our analysis. $\blacksquare$

3 Problem formulation

The optimization problem of interest is given by

[TABLE]

where the $a_{ij}$’s are positive scalar weights, the set $\mathcal{E}$ is the edge-set of a connected directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, and the matrices in the collection $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ belong to $O(d)$. The objective function $f_{1}$ comprises the weighted element-wise sum of squared differences between the $R_{ij}$-matrices and the $(R_{i}^{T}R_{j})$-matrices. The problem $(P_{1})$ is similar to the problem in [3]. The differences are that we allow for directed graphs (instead of undirected graphs) and we do not require the matrices to be contained in $SO(d)$.

The overall problem addressed in this paper is how to design distributed methods that achieve good solutions to $(P_{1})$.

When the $R_{i}$-matrices are orthogonal, $f_{1}$ can be written as

[TABLE]

The matrix $U_{1}$ will be used frequently in the following. The presented definition of $U_{1}$ might seem overly complicated, since $R_{i}^{-T}=R_{i}$ when $R_{i}$ is orthogonal. However, we will also use $U_{1}$ when the $R_{i}$-matrices are not orthogonal. In that case it is important to note however that $f_{1}$ is given by the definition in $(P_{1})$ and not by (11).

The matrix $L_{\text{undir}}$ is defined as

[TABLE]

where $[W]_{ij}=\begin{cases}a_{ij}R_{ij}&\text{ if }(i,j)\in\mathcal{E},\\ 0&\text{ else, }\end{cases},\>[A]_{ij}=\begin{cases}a_{ij}&\text{ if }(i,j)\in\mathcal{E},\\ 0&\text{ else, }\end{cases}$ $[\bar{W}]_{ij}=\begin{cases}\sqrt{a_{ij}}R_{ij}&\text{ if }(i,j)\in\mathcal{E},\\ 0&\text{ else, }\end{cases}$ and the operator $\text{diag}(\cdot)$ in the second term is understood in the block-matrix sense, i.e. $\text{diag}(W^{T}W)=(I_{n}\otimes 1_{d}1_{d}^{T})\odot W^{T}W$, where $\odot$ denotes element-wise multiplication. The matrix $A$ is now, compared to Section 2, a weighted adjacency matrix of $\mathcal{G}$. In the following, $A$ will always be defined in this way. The matrices $L_{\text{undir}}$, $W$, and $U_{1}$ are de-facto functions of the graph $\mathcal{G}$, $A$, and either the $R_{ij}$-matrices or the $R_{i}$-matrices. However, unless it is absolutely necessary, we will not show this dependence explicitly.

4 Symmetric Graphs

In this section we introduce Algorithm 1. It is the proposed distributed algorithm for synchronization over symmetric graphs. A detailed analysis of Algorithm 1 will be conducted in Section 4.2.

4.1 The algorithm

There are four matrices that can be seen as the output of the algorithm at each iteration. Those are: $\tilde{R}_{i}(k)$, ${R}_{i}(k)$, $\tilde{Q}_{i}(k)$, and ${Q}_{i}(k)$. The procedure to calculate the $\tilde{R}_{i}(k)$-matrices is similar to a gradient descent procedure and can also be seen as the power method. The ${R}_{i}(k)$-matrices are the projections of the $\tilde{R}_{i}^{T}(k)$-matrices onto $O(d)$.

For all $i$, the matrix $\tilde{Q}_{i}(k)$ and the corresponding ${Q}_{i}(k)$ is calculated from auxiliary variables. The most important such auxiliary variables are the $d_{is}(k)$’s, which are calculated in a distributed manner. The protocol for calculating the $d_{is}(k)$’s is similar to a well-known average consensus protocol, but differs by an extra term $(\tilde{d}_{is}(k-1)-\tilde{d}_{is}(k-2))$. This extra term makes the states converge not to the averages of the initial conditions, but to the averages over $j$ for the converging sequences $\{\tilde{d}_{js}(k)\}_{k=1}^{\infty}$. The idea behind the $\tilde{Q}_{i}(k)$-matrices is to modify the $\tilde{R}_{i}(k)$-matrices in such a way that the modified matrices (the $\tilde{Q}_{i}(k)$’s) converge to the matrices in the optimal solution to a spectral relaxation of problem ($P_{1}$) (this relaxation is $(P_{2})$ defined in Section 4.2.2).   Algorithm 1 Distributed method for symmetric graphs  Inputs: a symmetric directed graph $\mathcal{G}=(\mathcal{E},\mathcal{V})$, a weight matrix $A=[a_{ij}]$, and a collection $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $O(d)$.

Outputs: $\tilde{R}_{i}(k)$, ${R}_{i}(k)$, $\tilde{Q}_{i}(k)$, ${Q}_{i}(k)$ for $i\in\mathcal{V}$ and $k\geq 1$.

Initialization: let $\tilde{R}_{i}(0)=I_{d}$, ${d}_{is}(0)=1$, $\tilde{d}_{is}(0)=1$, and $\tilde{d}_{is}(-1)=1$ for all $i,s$. Let $V_{ij}=(a_{ij}+a_{ji})I$ and $Q_{ij}=a_{ij}R_{ij}+a_{ji}R_{ji}^{T}$ for all $(i,j)\in\mathcal{E}$. Let $\epsilon_{1},\epsilon_{2}>0$.

Iteration $k\geq 1$:

for all $i$, let

[TABLE]

where $\text{Pr}_{O(d)}$ is the projection operator (the least squares projection onto $O(d)$ computed by means of Singular Value Decomposition (SVD)).

The $d_{is}$-variables provide a way of creating the $\tilde{Q}_{i}(k)$-matrices by re-scaling the columns of the $\tilde{R}_{i}(k)$-matrices. This re-scaling is necessary to obtain the desired convergence. The ${Q}_{i}(k)$-matrices are projections onto $O(d)$ of scaled versions of the $\tilde{Q}_{i}^{T}(k)$-matrices. Under general conditions, those converge to the projections of the matrices in the optimal solution to the spectral relaxation of problem ($P_{1}$).

Subroutine 1 Calculation of $\tilde{Q}_{i}(k)$ and $\tilde{d}_{is}(k)$

Inputs: $\tilde{R}_{i}(k)$.

Outputs: $[\tilde{d}_{i1}(k),\tilde{d}_{i2}(k),\ldots,\tilde{d}_{in}(k)]^{T}$, $\tilde{Q}_{i}(k)$.

If $\tilde{R}_{i}(k)$ is not invertible or $\tilde{R}_{i}(k)$ is invertible and it does not hold that the eigenvalues of $\tilde{R}_{i}(k)^{-1}\tilde{R}_{i}(k-1)$ are distinct, real, and positive. Let $\tilde{Q}_{i}(k)=\tilde{R}_{i}(k)$ and $\tilde{d}_{is}(k)=1$ for all $s$. 2. 2.

Else using eigenvalue decomposition, compute

[TABLE]

Let $\tilde{Q}_{i}(k)=\tilde{R}_{i}(k)P_{i}^{-1}(k)(\bar{D}_{i}(k))^{-k}$.

Let $\tilde{d}_{is}(k)=\|\tilde{q}_{is}(k)\|^{2}_{2}\text{ for all }s,$ where $\tilde{Q}_{i}(k)=[\tilde{q}_{i1}(k),\tilde{q}_{i2}(k),\ldots,\tilde{q}_{id}(k)]$ and each $\tilde{q}_{is}\in\mathbb{R}^{d}$.

Let $\tilde{R}(k)=[\tilde{R}_{1}^{T}(k),\tilde{R}_{2}^{T}(k),\ldots,\tilde{R}_{n}^{T}(k)]^{T}$ for all $k$. The update for $\tilde{R}(k)$ is given by

[TABLE]

4.2 Analysis

In this section we show how the matrices $\tilde{R}_{i}(k)$, ${R}_{i}(k)$, $\tilde{Q}_{i}(k)$, and ${Q}_{i}(k)$ relate to problem $(P_{1})$. We will provide conditions for well-posedness and convergence.

4.2.1 Some properties of $L_{\text{undir}}$

For the analysis of Algorithm 1 we first provide an alternative definition of transitive consistency, formulated in terms of the $L_{\text{undir}}$-matrix. To be more precise, in Proposition 8 we state that for the general case of invertible matrices, transitive consistency is equivalent to the $L_{\text{undir}}$-matrix having a $d$-dimensional nullspace. In other words, there are no collections of matrices that are not transitively consistent for which the $L_{\text{undir}}$-matrix has a nullspace of dimension $d$. This motivates the choice of $f_{1}$ as the objective function in an optimization problem for synchronization of matrices.

Proposition 8**.**

For collections $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$ and graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ that is connected, it holds that

[TABLE]

with equality if and only if transitive consistency holds.

Before we provide the proof of Proposition 8 we provide the following lemma and the proof thereof.

Lemma 9**.**

For any connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$, the collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent for $\mathcal{G}$ if and only if there is a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ such that

[TABLE]

*Proof: * Suppose $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent, then, according to Lemma 7, there is $\{R_{i}\}_{i\in\mathcal{V}}$ such that (10) holds for the $R_{i}$-matrices. In this case it holds that

[TABLE]

which implies that (17) is fulfilled since $L_{\text{undir}}$ is symmetric. On the other hand, if $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ is not transitively consistent, there are no $R_{i}$ such that (10) holds. It can now be shown that (18) does not hold for any collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$. $\blacksquare$

*Proof of Proposition 8: *

Part 1: Here we assume that $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent. Due to Lemma 9, we know that

[TABLE]

Thus we need to show that the inequality in (19) cannot be strict. Since $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent, there is $\{R_{i}\}_{i\in\mathcal{V}}$, where the $R_{i}$ fulfill (10).

Suppose the inequality (19) is strict. We know that $\text{im}(U_{1})\subset\text{ker}(L_{\text{undir}}).$ Now there must be a vector $y=[y_{1}^{T},y_{2}^{T},\ldots,y_{n}^{T}]^{T}\in\mathbb{R}^{nd}$, where the $y_{i}$ are in $\mathbb{R}^{d}$, such that $y\in\text{ker}(L_{\text{undir}}),$ $y\neq 0$, and $y^{T}U_{1}=0$. There must be $k$ and $l$ such that the $l$-th element of $y_{k}$ is nonzero. Now, let

[TABLE]

and $\bar{Y}=[\bar{x}_{1},\bar{x}_{2},\ldots,\bar{x}_{l-1},y,\bar{x}_{l+1},\bar{x}_{d}],$ where $\bar{x}_{s}\in\mathbb{R}^{nd}$ for $s=1,2,\ldots n$. It holds that $y^{T}\bar{X}=0$ and $L_{\text{undir}}\bar{Y}=0$. For all $i$, let $\bar{Y}_{i}$ denote the $i$-th $d\times d$ block matrix in $\bar{Y}$. The rest of this part of the proof consists of firstly showing that all the $\bar{Y}_{i}$-matrices are invertible and secondly showing that we can use those matrices to formulate a contradictory statement.

It holds that $\bar{Y}_{k}\in GL(d,\mathbb{R})$. This is true since it is constructed by taking the identity matrix and replacing the $l$-th column by another vector that has a nonzero $l$-th element. Now, for any $j\in\mathcal{N}_{k}$ it holds that ${\|R_{kj}\bar{Y}_{j}-\bar{Y}_{k}\|_{F}=0,}$ which implies that $\bar{Y}_{j}\in GL(d,\mathbb{R})$. Also, for any $i$ such that $k\in\mathcal{N}_{i}$, it holds that $\|R_{ik}\bar{Y}_{k}-\bar{Y}_{i}\|_{F}=0,$ which implies that $\bar{Y}_{i}\in GL(d,\mathbb{R})$. Now, due to the fact that $\mathcal{G}$ is connected, an induction argument can be used to show that all the $\bar{Y}_{i}$ are elements in $GL(d,\mathbb{R})$.

The collection $\{\bar{Y}_{i}\}_{i\in\mathcal{V}}$ satisfies $R_{ij}=\bar{Y}_{i}\bar{Y}_{j}^{-1}$ for all $(i,j)\in\mathcal{E}.$ Since $y\not\in\text{im}(U_{1})$, the two collections $\{\bar{Y}^{-1}_{i}\}_{i\in\mathcal{V}}$ and $\{R_{i}\}_{i\in\mathcal{V}}$ are not equal up to transformation from the left. But, since the graph is connected, the two must be equal up to transformation from the left (Lemma 7). This is a contradiction. Hence it is a false assumption that the inequality in (19) is strict.

Part 2: Here we show that if ${\text{dim}(\text{ker}(L_{\text{undir}}}))\geq d,$ then $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent.

Let $\bar{Y}=[y_{1},y_{2},\ldots,y_{nd}]^{T}\in\mathbb{R}^{nd\times d}$ be any full rank matrix such that $L_{\text{undir}}\bar{Y}=0.$ It holds that all the $y_{i}\in\mathbb{R}^{d}$. Let $\bar{Y}_{i}$ be the $i$-th $d\times d$ block matrix in $\bar{Y}$. Since $\bar{Y}$ is full rank, there is a collection $\{y_{i_{j}}\}_{j=1}^{d}$ such that $[y_{i_{1}},y_{i_{2}},\ldots y_{i_{d}}]\in GL(d,\mathbb{R})$.

Now, for $k\in\mathcal{V}$ we know that for any $j\in\mathcal{N}_{k}$ it holds that $\|R_{kj}\bar{Y}_{j}-\bar{Y}_{k}\|_{F}=0,$ which implies that $\text{im}(\bar{Y}_{j}^{T})=\text{im}(\bar{Y}_{k}^{T})$. Also, for any $i$ such that $k\in\mathcal{N}_{i}$, it holds that

[TABLE]

which implies that $\text{im}(\bar{Y}_{i}^{T})=\text{im}(\bar{Y}_{k}^{T})$. Now, due to the fact that $\mathcal{G}$ is connected, an induction argument can be used to show that $\text{im}(\bar{Y}_{j}^{T})=\text{im}(\bar{Y}_{i}^{T})$ for all $i,j$. But then $\text{im}([y_{i_{1}},y_{i_{2}},\ldots y_{i_{d}}])\subset\text{im}(\bar{Y}_{j}^{T})\text{ for all }j,$ which together with the fact that $[y_{i_{1}},y_{i_{2}},\ldots y_{i_{d}}]\in GL(d,\mathbb{R})$ is full rank, implies that $\bar{Y}_{i}\in GL(d,\mathbb{R})$ for all $i$. It holds that $\text{im}(U_{1}(\{\bar{Y}^{-1}_{i}\}_{i\in\mathcal{V}}))\subset\text{ker}(L_{\text{undir}}).$ Now the desired result follows by application of Lemma 9. $\blacksquare$

From Definition 5 and Proposition 8 we get the following equivalent characterisations of transitive consistency.

Corollary 10**.**

*(equivalent characterisations of transitive consistency)

For a connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and a collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$ the following three statements are equivalent*

$\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$* is transitively consistent. * 2. 2.

*There is a collection $\{\bar{R}_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ such that $\text{ker}(L_{\text{undir}}(\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}))=\text{im}(U_{1}(\{\bar{R}_{i}\}_{i\in\mathcal{V}}))$. * 3. 3.

There is a collection $\{\bar{R}_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ such that $R_{ij}=\bar{R}^{-1}_{i}\bar{R}_{j}\text{ for all }(i,j)\in\mathcal{E}.$

According to Corollary 10, the following holds. For a collection $\{\bar{R}_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$, $f_{1}$ attains the value [math] if and only if the collection $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent. This means that minimization of the right-hand side of (11) is an approach to consider even when the $R_{ij}$ are not necessarily orthogonal. This is the approach in the first step of an iterative method recently published [33].

With the assurance given by Proposition 8 that $f_{1}$ is a suitable objective function, we now move on to the convergence analysis of Algorithm 1.

4.2.2 Convergence analysis

We begin by introducing a relaxation of problem $(P_{1})$, given by

[TABLE]

Let $\bar{X}=[\bar{R}_{1}^{T},\bar{R}_{2}^{T},\ldots,\bar{R}_{n}^{T}]^{T}$ be one of the optimal solutions to $(P_{2})$. Under the assumption that $R_{ij}=R_{ji}^{T}$ for all $(i,j)\in\mathcal{E}$, the spectral relaxation method in Section 2.2 in [3] is the same as solving problem $(P_{2})$.

Now we provide a list of conditions for convergence, which are recalled in the following propositions. Only a subset of the conditions will be used in each proposition.

The conditions (1-3) are fundamental properties that need no further explanation. Conditions (4-5) and (10) are conditions for the step size determination. These have the property that they scale with the number of nodes in the network, i.e., $n$. Condition (6) states that all the $\bar{R}_{i}$-matrices in the optimal solution to $(P_{2})$ are invertible and condition (7) states that the sum of those is invertible. Condition (8) states that the $d$ smallest eigenvalues of $L_{\text{undir}}$ are strictly smaller than the $(n-1)d$ largest. Condition (9) states that the $d$ smallest eigenvalues of $L_{\text{undir}}$ are distinct.

The following lemma provides a bound for $\epsilon_{1}$ such that the discrete-time system defined by (15) is stable. It is a justification of convergence condition (4).

Lemma 11**.**

The largest eigenvalue of $P$, as defined in convergence condition (4), is an upper bound for the eigenvalues of $L_{\text{undir}}$ for all graphs $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and collections $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ satisfying convergence conditions (1) and (2).

*Proof: * The largest eigenvalue $\lambda_{\max}$ is given by

[TABLE]

where $\mathbb{S}^{nd}$ is the $nd$-dimensional unit sphere. Let $v=[v_{1}^{T},v_{2}^{T},\ldots,v_{n}^{T}]^{T}\in\mathbb{S}^{nd}$, where each $v_{i}\in\mathbb{R}^{d}$. By using the structure of the $f_{1}$-function in $(P_{1})$, one can show that

$v^{T}L_{\text{undir}}v=\sum\limits_{(i,j)\in\mathcal{E}}a_{ij}\|R_{ij}v_{j}-v_{i}\|_{2}^{2}.$ Now, $v^{T}L_{\text{undir}}v\leq\sum\limits_{(i,j)\in\mathcal{E}}a_{ij}(\|v_{i}\|_{2}+\|v_{j}\|_{2})^{2}\leq\max_{u\in\mathbb{S}^{n}\cap(\mathbb{R}^{+})^{n}}u^{T}Pu.$ The set $\mathbb{R}^{+}$ comprises the non-negative real numbers. $\blacksquare$

Lemma 11 has the following implication. If $\epsilon_{1}$ in Algorithm 1 is chosen to be smaller than $2/\|P\|_{2}$, then $\tilde{R}(k)$ (see (15)) converges as $k$ goes to infinity.

Now, unless the $R_{ij}$-matrices are transitively consistent, the nullspace of $L_{\text{undir}}$ has a lower dimension than $d$ and in general it will be zero-dimensional. Thus, $\tilde{R}(k)$ converges to zero. In the case when the $R_{ij}$-matrices are transitively consistent, the $\tilde{R}(k)$ converge to a $d$-dimensional subspace in general.

Now we provide a result for the special case when transitive consistency holds. Algorithm 1 reduces to the first two lines in each iteration, which is de facto the power method. See [28] for a discussion about the power method in a similar context. We provide Proposition 12 and its proof below for the sake of completeness.

Proposition 12**.**

Suppose convergence conditions (1-3), and (5-7) hold. Then, for Algorithm 1, there is a positive integer $K$ such that $\tilde{R}_{j}^{-1}(k)$ is well defined for $k\geq K$, and (for $k\geq K$) it holds that

[TABLE]

*Proof: * Under conditions (2) and (3), it holds that $L_{\text{undir}}$ is similar to the matrix $\bar{L}\otimes I_{d}$, where $\bar{L}$ is defined in convergence condition (5). This is a consequence of the fact that we can write $L_{\text{undir}}$ as

[TABLE]

In the right-hand side above, $\text{diag}(\cdot)$ shall be interpreted in the block diagonal sense, where the $R_{i}^{T}$ and $R_{i}$ are put as blocks on the diagonal.

Since conditions (1), (2), and (3) hold, we can use Corollary 10. There is a collection $\{\bar{R}_{i}\}_{i\in\mathcal{V}}$ such that the conditions (1) and (2) are fulfilled (in the corollary). Thus, $\text{ker}(L_{\text{undir}})=\text{im}(U_{1}(\{\bar{R}_{i}\}_{i\in\mathcal{V}})).$ Due to this fact and the fact that $\tilde{R}_{i}(0)=I_{d}$ for all $i$, it holds that the projection of $\tilde{R}^{T}(0)$ onto $\text{ker}(L_{\text{undir}})$ is $U_{1}U_{1}^{T}\tilde{R}(0)=U_{1}V,$ where $V=\frac{1}{n}\sum_{i\in\mathcal{V}}\bar{R}_{i}$. Furthermore, $R(k)$ converges to $U_{1}(\{\bar{R}_{i}\}_{i\in\mathcal{V}})V$ as $k$ goes to infinity (condition (5)). Now, if $V$ is invertible (condition (7)), by the definition of the limit there is a $K$ such that $\tilde{R}_{j}^{-1}(k)$ is well defined for all $k\geq K$. It holds that

[TABLE]

Convergence conditions (6) and (7) holds. Thus, the $(\bar{R}_{i}V)$-matrices are invertible. Under these conditions, the projections of $\tilde{R}_{i}(k)$ converge in (21) (see the last paragraph of Proposition 14 below for details about the convergence of the projections). $\blacksquare$

In Proposition 12 it is important to note that for all $(i,j)\in\mathcal{E}$, $R_{i}^{T}(k)R_{j}(k)$ converges to $R_{ij}\in{O}(d)$ inside $O(d)$, whereas $\tilde{R}_{i}(k)\tilde{R}_{j}^{T}(k)$ converges to $R_{ij}\in{O}(d)$ without guarantees of being in $O(d)$ for each $k$.

Now we take a step further in our analysis of Algorithm 1. We show that when the $R_{ij}$-matrices are not necessarily transitively consistent, we still have a nice convergence property for the $\tilde{R}_{i}(k)$-matrices.

Proposition 13**.**

Suppose that the convergence conditions (1-2), (4), and (6-8) are satisfied. Then, for Algorithm 1 there is a positive integer $K$ such that $\tilde{R}_{j}^{-1}(k)$ is well defined for all $k\geq K$, and (for $k\geq K$) it holds that

[TABLE]

*Proof: * Since the convergence conditions (1), (2), and (4) are fulfilled, we know that the discrete time system defined in equation (15) is stable.

The columns of $\bar{X}$ are, up to scale and orthogonal transformation, the eigenvectors corresponding to the $d$ smallest eigenvalues of $L_{\text{dir}}$. Now, let us rewrite $\tilde{R}(k)$ as $\tilde{R}(k)=VV_{1}(k)V_{1}+(1/\sqrt{n})\bar{X}V_{2}(k)V_{2},$ where the columns of the matrix $V$ are the eigenvectors corresponding to the $(n-1)d$ largest eigenvalues of the matrix $L_{\text{undir}}$. Due to convergence condition (8), there are $(n-1)d$ eigenvalues that are strictly larger than the other $d$ eigenvalues. The matrices $V_{1}(k)$ and $V_{2}(k)$ are defined as

[TABLE]

and the matrices $V_{1}$ and $V_{2}$ are defined as $V_{1}=V^{T}\tilde{R}(0)$ and $V_{2}=\frac{1}{\sqrt{n}}\bar{X}^{T}\tilde{R}(0)=\frac{1}{n}\sum_{i\in\mathcal{V}}\bar{R}_{i}$.

Now, by using the definitions of $V_{1}(k)$ and $V_{2}(k)$ together with the fact that convergence condition (8) holds, we can formulate the following convergence result:

[TABLE]

We also see from the definition of $V_{2}(k)$, that $V_{2}(k)$ is invertible for all $k$. For any pair $(i,j)\in\mathcal{E}$, if $\tilde{R}_{j}(k)$ is invertible, it holds that

[TABLE]

and $\tilde{R}_{j}(k)$ is invertible if and only if $\tilde{R}_{j}(k)V_{1}^{-1}(k)$ is.

For any $i$ it holds that

[TABLE]

The matrix $(V_{2}(k)V_{2}V_{2}(k)^{-1})$ is similar to $V_{2}$ for all $k$, i.e., the eigenvalues are the same for the two. This means that the norm of $((1/\sqrt{n})\bar{R}_{i}V_{2}(k)V_{2}V_{2}(k)^{-1})$ is bounded from below by some positive constant. Now we consider the second term in the sum on the right-hand side of (27). Each element in that term is equal to a linear combination of rational expressions $([V_{1}(k)]_{jj}/[V_{2}(k)]_{ll})$ for $j,l\in\{1,2,\ldots,d\}$. According to (25), all those go to zero as $k$ goes to infinity.

Now, if $k$ is large enough, $\rho(k)$ is small enough for all the $\tilde{R}_{i}(k)(V_{2}(k))^{-1}$, and equivalently, $\tilde{R}_{i}(k)$ to be invertible. For such $k$, it holds that

[TABLE]

for all $(i,j)\in\mathcal{E}$. Thus, $\tilde{R}_{i}(k)\tilde{R}_{j}^{-1}(k)\rightarrow\bar{R}_{i}\bar{R}_{j}^{-1}$ as $k\rightarrow\infty$. $\blacksquare$

Before we proceed, we introduce $\mathcal{S}_{(P_{2})}=\{\bar{X}Q:Q\in O(d)\}.$ When the convergence conditions (1), (2), and (8) hold, $\mathcal{S}_{(P_{2})}$ comprises the entire set of solutions to problem $(P_{2})$.

So far we have only addressed the convergence properties for $\tilde{R}_{i}(k)$ and $R_{i}(k)$. Now we analyze the convergence of $\tilde{Q}_{i}(k)$ and $Q_{i}(k)$. The following result is a main result of this work. In the case when the $R_{ij}$-matrices are not necessarily transitively consistent it guarantees convergence to the projected matrices obtained from the optimal solution to problem $(P_{2})$. It can be seen as an equivalent result to that of Proposition 12 for the $R_{i}(k)$-matrices, but this time formulated for the $Q_{i}(k)$-matrices for the case when transitive consistency does not necessarily hold for the $R_{ij}$-matrices.

Proposition 14**.**

Suppose that the convergence conditions (1-2), (4), and (6-10) are satisfied. Then, for Algorithm 1 there is a positive integer $K$ such that $\tilde{R}_{i}^{-1}(k)$ is well defined for all $i$ and $k\geq K$, and (for $k\geq K$) it holds that

[TABLE]

and

[TABLE]

*Proof: * The proof will use the notation from Proposition 13. According to Proposition 13 there is a $K$ such that $\tilde{R}_{j}^{-1}(k)$ is well defined for all $k\geq K-1$. In the following we assume that $k\geq K-1$.

[TABLE]

where the right-hand side is the Jordan decomposition. It holds that

[TABLE]

and

[TABLE]

Now, similar to (27) we see that each right-hand side of (30) and (31) a sum of two terms. The first terms have bounded eigenvalues and the second terms are $\mathcal{O}(\rho(k))$ and $\mathcal{O}(\rho(k-1))$, respectively. Thus, for all $i$ it holds that

[TABLE]

which means, due to (25), that for all $i$ it holds that

[TABLE]

Now the following holds:

[TABLE]

where $\bar{Z}$ is a diagonal matrix with real nonzero elements on the diagonal. Equation (34) is a consequence of the continuity of the eigenvalues. For equation (35) the key point is that convergence property (9) holds. Thus, the diagonal elements of $V_{2}(1)$ are distinct and real. Then it holds that for $k$ large enough, the elements of ${D}_{i}(k)$ are distinct and real. This guarantees the convergence of $P_{i}(k)$, see for example Lemma 12 in [34].

The matrix $\tilde{Q}_{i}(k)$ is defined as

[TABLE]

It holds that

[TABLE]

We will now prove that

[TABLE]

To prove (37), we prove that

[TABLE]

The expressions in (38) can be manipulated as follows.

[TABLE]

In these derivations we have first used the fact that

[TABLE]

Then we have used (25), (32), (35), the fact that $V_{2}(1)^{-k}=V_{2}^{-1}(k)$, and the fact that the elements of $V_{2}(k)$ are bounded by $1$ in magnitude. Now, due to the derivations above we can use (25) and (35) to conclude that the convergences in (38) hold.

The matrix $D_{i}(k)$ is defined as

[TABLE]

for all $i$. We will now prove that

[TABLE]

We recall that $\tilde{d}_{is}(k)=\|\tilde{q}_{is}(k)\|^{2}_{2}\text{ for all }s,$ where $\tilde{Q}_{i}(k)=[\tilde{q}_{i1}(k),\tilde{q}_{i2}(k),\ldots,\tilde{q}_{id}(k)]$ and each $\tilde{q}_{is}\in\mathbb{R}^{d}$. We see from (37) that $\tilde{Q}_{i}(k)$ converges to a constant matrix for all $i$. This means that the $\tilde{d}_{is}$ converge to constants and the averages of those constants are equal to the corresponding elements of $\frac{1}{n}[\bar{Z}^{-2}]_{ss}$. More precisely,

[TABLE]

We will show that for all $i$ and all $s$ it holds that

[TABLE]

under the protocol

[TABLE]

and the initial conditions chosen as in Algorithm 1. The protocol is equal to an average consensus protocol up to the extra term $(\tilde{d}_{is}(k)-\tilde{d}_{is}(k-1))$. Without the extra term it is well known that the $d_{is}(k)$’s converge to the averages of the initial conditions under convergence condition (10), see for example [26]. With the term and convergence condition (10), the convergence in (41) holds. Now we show this.

We can rewrite (43) as

[TABLE]

where $\bar{\xi}_{s}(k)=\frac{1}{n}\sum_{l=1}^{n}\xi_{ls}(k)$, $\bar{\xi}_{is}(k)=\frac{1}{n}\sum_{l=1}^{n}(\xi_{is}(k)-\xi_{ls}(k))$, and $\xi_{is}(k)=(\tilde{d}_{is}(k)-\tilde{d}_{is}(k-1))$. The matrix $L$ is the graph Laplacian matrix of the graph $\mathcal{G}$ (see Section 2) and the matrix $(I_{d}-\epsilon_{2}L)$ is a stochastic matrix (convergence condition (10) guarantees that the matrix is stochastic, see [26]). Since we have chosen $d_{is}(0)=0$, $\tilde{d}_{is}(0)=\tilde{d}_{is}(-1)=1$ for all $i,s$ it holds that

[TABLE]

The right-hand side of (45) is written as the sum of two parts. We are done with this section of the proof if it holds that the second part goes to zero as $k$ goes to infinity. Let $P$ be the orthogonal matrix on the left-hand side of in the spectral factorization of $(I_{d}-\epsilon_{2}L)$. We can write $P=[P_{1},P_{2}]$, where $P_{1}$ is a matrix containing the eigenvectors corresponding to eigenvalues in the interior of the unit disc and $P_{2}$ is the vector corresponding to the eigenvalue that is $1$. We know that such a decomposition exists, see convergence condition (1). It holds that $P_{2}^{T}[\bar{\xi}_{1s}(k),\bar{\xi}_{2s}(k),\ldots,\bar{\xi}_{ns}(k)]^{T}=0$ for all $k$ and $\|[\bar{\xi}_{1s}(k),\bar{\xi}_{2s}(k),\ldots,\bar{\xi}_{ns}(k)]^{T}\|$ goes to zero as $k$ goes to infinity. Thus, after a coordinate transformation with $P$, the second part on the right-hand side of (45) is equivalent to $n$ convolutions between the impulse responses of stable systems and bounded inputs that converge to zero in norm. Hence the second part goes to zero as $k$ goes to infinity.

Now, by using (37), (41), and (42) we deduce that

[TABLE]

where $\bar{P}$ is an orthogonal matrix. By definition it holds that

[TABLE]

$\text{Pr}_{{O}(d)}$ is defined by $\text{Pr}_{\mathit{O}(d)}(W)=W_{1}W_{2}^{T}$, where $(W_{1},S,W_{2})$ is the Singular Value Decomposition (SVD) of $W\in\mathbb{R}^{d\times d}$. We refer to $W_{1}$ as the “left matrix” and $W_{2}$ as the “right matrix”. The matrix $\bar{R}_{i}\bar{P}$ is invertible (convergence condition (6)), and for $k$ large enough the sign of the determinant of $(\tilde{Q}_{i}(k)D_{i}^{-\frac{1}{2}}(k))$ equals that of $\bar{R}_{i}\bar{P}$. This means that for $k$ large enough the projections will be either only onto $SO(d)$ or either only onto $O(d)-SO(d)$. Up to sign the left respective right matrices in the SVD of $(\tilde{Q}_{i}(k)D_{i}^{-\frac{1}{2}}(k))$ converge to the left respective the right matrix in the SVD of $\bar{R}_{i}\bar{P}$. When projection is performed, these possible sign differences cancel out in the product between the left and the right matrices. Thus in (47), the expression on the right-hand side goes to $\text{Pr}_{\mathcal{O}(d)}(\bar{R}_{i}\bar{P})=\text{Pr}_{\mathcal{O}(d)}(\bar{R}_{i})\bar{P}$ $\text{as }k\rightarrow\infty.$ But since $\bar{P}$ is orthogonal, it holds that

[TABLE]

$\blacksquare$

The procedure for the $d_{ij}$’s, i.e., to distributed computation of a time-varying signal is equivalent to that in [35].

4.3 Numerical simulations

For the collections $\{R_{i}\}_{i\in\mathcal{V}}$ and $\{\tilde{R}_{i}^{-1}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ we define

[TABLE]

For the gap-function to be well-defined, we also assume that the graph $\mathcal{G}$ is connected and that the $R_{ij}$-matrices in problem $(P_{1})$ are not transitively consistent. We know according to Proposition 13 and Proposition 14 that

[TABLE]

as $k\rightarrow\infty$, where the $\bar{R}_{i}$-matrices are the matrix blocks in the optimal solution $\bar{X}$ to problem $(P_{2})$. By plotting the logarithm of the gap as a function of $k$, Fig. 1 shows that this convergence holds in numerical simulations. We will now explain this figure in more detail.

The left two figures show the evolution of

[TABLE]

for two different parameter settings. The right two figures show the evolution of

[TABLE]

for two different parameter settings. The means over 100 simulations are shown. For each simulation $n=10$ and $d=5$; the $R_{ij}$-matrices are created in a two step procedure. First $R_{i}$-matrices are drawn from the uniform distribution on $O(5)$. Then element-wise Gaussian noise is added with standard deviation $0.2$ to the $R_{i}^{T}R_{j}$-matrices. Those noisy matrices are then projected back onto $O(d)$ to obtain the $R_{ij}$-matrices. All the $a_{ij}$ are either equal to $1$ or [math]; the graph $\mathcal{G}$ is chosen randomly from the set of connected graphs with a certain graph density (percentage of edges). In the top figures the graph density is chosen to $0.9$ in each simulation, whereas in the bottom figures the graph density is chosen to $0.6$ in each simulation. The step sizes $\epsilon_{1}$ and $\epsilon_{2}$ are both chosen to $1/(2n)$, which is an upper bound for $\frac{2}{\|P\|_{2}}$.

We see that the convergence is faster when the connectivity in the graph, i.e., graph density, is stronger. We also see that the performance of the $R_{i}(k)$-matrices is better initially than that of the $Q_{i}(k)$-matrices, but after sufficiently many iterations the performance of the $Q_{i}(k)$-matrices is several orders of magnitude better (top left and bottom left figures). This out-performance is to be expected since the gap is proven to converge to zero for the $Q_{i}(k)$-matrices (see Proposition 14) but not for the $R_{i}(k)$-matrices.

5 Directed graphs

In this section we introduce a distributed algorithm for the synchronization problem over connected graphs that are directed and possibly asymmetric. Such graphs are not necessarily strongly connected. The asymmetric communication scenario is present when some agents are not able to transmit information due to for example malfunctioning equipment. Asymmetric communication could also be deliberately imposed. For example if one wants a hierarchical communication structure.

5.1 A problem formulation for directed graphs

We begin this section by introducing the following problem

[TABLE]

It is obviously not the same problem as $(P_{1})$. However, it has a structure that allows for the development of a distributed algorithm in situations where the graph could be asymmetric. The idea is that the optimal solution to this problem should be close to that of problem $(P_{1})$. When we design an algorithm in this section the objective of the algorithm will be to calculate a good feasible solution to $(P_{3})$.

We will now explain some connections between the two problems $(P_{1})$ and $(P_{3})$. The cost function $f_{2}$ can be written as

[TABLE]

where

[TABLE]

It holds that

[TABLE]

where diag in the second term on the right-hand side is defined in same way as for the second term in (13).

Let $\bar{L}_{\text{dir}}$ be equivalent to the matrix ${L}_{\text{dir}}$ defined for the graph $\bar{\mathcal{G}}$ and the matrices $\{\bar{{R}}_{ij}\}_{(i,j)\in\bar{\mathcal{E}}}$, where $\bar{R}_{ij}=R_{ji}^{T}$. It holds that

[TABLE]

If the graph $\mathcal{G}$ is symmetric and $A=A^{T}$, it holds that

[TABLE]

Furthermore, if it also holds that $R_{ij}^{T}=R_{ji}$ for all $i$, $j$, then

[TABLE]

5.2 The algorithm

Algorithm 2 Distributed method for graphs that could be asymmetric  Inputs: a directed graph $\mathcal{G}=(\mathcal{E},\mathcal{V})$, a weight matrix $A=[a_{ij}]$, and a collection $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $O(d)$.

Outputs: $\tilde{R}_{i}(k)$, ${R}_{i}(k)$ for $i\in\mathcal{V}$ and $k\geq 1$.

Initialization: let $\tilde{R}_{i}(0)=I_{d}$ for all $i$ and $\epsilon_{3}>0$.

Iteration $k\geq 1$:

for all $i$ and all $k\geq 1$ we define $\tilde{R}_{i}(k)$ via the following equations.

[TABLE]

Let $\tilde{R}(k)=[\tilde{R}_{1}^{T}(k),\tilde{R}_{2}^{T}(k),\ldots,\tilde{R}_{n}^{T}(k)]$ for all $k$, the update for $\tilde{R}(k)$ is written as

[TABLE]

5.3 Analysis

The matrix $L_{\text{dir}}$ is defined for directed graphs (and asymmetric graphs). It is related to the graph connection Laplacian matrix [2, 3, 30], which however is only defined for undirected graphs.

5.3.1 Some properties of $L_{\text{dir}}$

The matrix $L_{\text{dir}}$ could be seen as a way of extending $L_{\text{undir}}$ to the case of asymmetric graphs, but the matrix is also defined for symmetric graphs. We can show a strong relation between $L_{\text{undir}}$ and synchronization/transitive consistency of invertible matrices, see Proposition 8. The equivalent relation does not hold when we consider $L_{\text{dir}}$. The question is what we can say about $L_{\text{dir}}$. The propositions 15, 18 and 19 provide some answers to this question.

Proposition 15 formulates a nullspace condition for $L_{\text{dir}}$ that must hold for transitively consistent matrices. This is a necessary condition, similar to that of Proposition 8, but under the stronger assumption that the graph is QSC. However, as stated in Proposition 18 further below, this condition is not a sufficient condition for synchronization of invertible matrices. Proposition 18 shows that there is an infinity of non-synchronized matrix collections close to a synchronized collection $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$, all satisfying the nullspace condition. Proposition 19 states that $(P_{3})$ has an optimal value equal to [math] if and only if the $R_{ij}$’s are synchronized. It shares similarities to Lemma 9 formulated for $L_{\text{undir}}$.

Before we proceed we define the matrix $\bar{L}=\text{diag}(A1_{n})-A$.

Proposition 15**.**

For any (QSC) graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$ that is transitively consistent for $\mathcal{G}$, and collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ it holds that

[TABLE]

(if and) only if

[TABLE]

for any matrix $V$, where the columns thereof form a basis for $\text{ker}(L_{\text{dir}})$. In particular, if $\mathcal{G}$ is QSC, (58) can be stated as

[TABLE]

*Proof: *

Only if: Suppose it holds that $R_{ij}=R^{-1}_{i}R_{j}\text{ for all }(i,j)\in\mathcal{E}.$ Then

[TABLE]

Now, $L_{\text{dir}}V=0\Leftrightarrow(\bar{L}\otimes I)\text{diag}(R_{1},R_{2},\ldots,R_{n})V=0\Leftrightarrow\text{im}(\text{diag}\left(R_{1},R_{2},\ldots,R_{n}\right)V)=\text{ker}(\bar{L}\otimes I_{d}).$

If: This part only concerns the case when the graph $\mathcal{G}$ is QSC.

Since $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent for $\mathcal{G}$, there is a collection $\{R^{**}_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$ such that

[TABLE]

Thus, the nullspace of $L_{\text{dir}}$ is given by $\text{ker}(L_{\text{dir}})=\text{im}(V),$ where $V=\text{diag}(R^{**-1}_{1},R^{**-1}_{2},\ldots,R^{**-1}_{n})([1,1,\ldots,1]^{T}\otimes I_{d}).$

Now, suppose (58) holds. Then $\text{diag}\left(R_{1},R_{2},\ldots,R_{n}\right)V=([1,1,\ldots,1]^{T}\otimes I_{d})Q,$ where $Q$ is some matrix in $GL(d,\mathbb{R})$. This means that

[TABLE]

which implies that $\{R^{**}_{i}\}_{i\in\mathcal{V}}$ and $\{R_{i}\}_{i\in\mathcal{V}}$ are equal up to transformation from the left. By using Lemma 7 we conclude that $R_{ij}=R^{-1}_{i}R_{j}\text{ for all }(i,j)\in\mathcal{E}.$ $\blacksquare$

Remark 16**.**

In Proposition 15, the relation

[TABLE]

holds if and only if for any matrix $V_{2}$, where the columns thereof comprise a basis for $\text{ker}(L\otimes I_{d})$, there is a matrix $Q$ such that

[TABLE]

Remark 17**.**

In Proposition 15, if $\mathcal{G}$ is connected but not QSC, it can hold that $A^{T}$ is the adjacency matrix of a QSC graph $\mathcal{G}^{\prime}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime})$. Then it holds that

[TABLE]

Proposition 15 is important as it states that for transitively consistent matrices and QSC graphs, the conditions (10) and (58) are equivalent. For invertible matrices it is not true that (58) implies transitive consistency. See Proposition 18 below. However, if we restrict all matrices to be orthogonal, the implication is true, see Proposition 19 below. The latter proposition guarantees that the problem $(P_{3})$ is well posed in the sense that the objective function is equal to zero if and only if the $R_{ij}$-matrices are synchronized.

Proposition 18**.**

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be any QSC graph such that the node degree of one node is at least 2 (i.e, at least one element in the vector $A(\mathcal{G})[1,1,\ldots,1]^{T}$ is greater or equal to 2). Let $\{{R}^{*}_{ij}\}_{(i,j)\in\mathcal{E}}$ be a collection of matrices in $GL(d,\mathbb{R})$ that is transitively consistent for $\mathcal{G}$. Let $\{R^{*}_{i}\}_{i\in\mathcal{V}}$ be a collection of matrices in $GL(d,\mathbb{R})$ for which it holds that

[TABLE]

Now, for any $\epsilon>0$, there is a collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $GL(d,\mathbb{R})$ that is not transitively consistent for $\mathcal{G}$ such that

[TABLE]

and (59) holds for $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ and a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $GL(d,\mathbb{R})$.

*Proof: *

Suppose the $k$-th element of the vector $A(\mathcal{G})[1,1,\ldots,1]^{T}$ is larger or equal to $2$. Then there is $l,m$ such that $l\neq k$, $m\neq k$, $R^{*}_{kl},R^{*}_{km}\in GL(d,\mathbb{R})$. Let $R_{kl}=R^{*-1}_{k}(I+Q)R^{*}_{l}$ and $R_{km}=R^{*-1}_{k}(I-Q)R_{m}^{*}$, where we choose $Q\neq 0$ such that $(I\pm Q)\in GL(d,\mathbb{R})$ and

[TABLE]

Furthermore, let $R_{ij}=R^{*}_{ij}$ for all $(i,j)\not\in\{(k,l),(k,m)\}$. (65) is satisfied. By construction, all the $R_{ij}$ are elements of $GL(d,\mathbb{R})$.

Let $R_{i}=R_{i}^{*}$ for all $i$. It holds that

[TABLE]

where $\mathbb{R}^{nd\times nd}\ni\bar{Q}=[Q_{ij}]$, $Q_{kl}=a_{kl}Q$, $Q_{km}=-a_{km}Q$ and $Q_{ij}=0\in\mathbb{R}^{d\times d}$ for all $(i,j)\not\in\{(k,l),(k,m)\}$. It holds that $\text{ker}((\bar{L}\otimes I)+\bar{Q})\supset\text{ker}(\bar{L}\otimes I)$. Since the eigenvalues of $L_{\text{dir}}(\mathcal{G},\{{R}_{ij}\}_{(i,j)\in\mathcal{E}})$ are continuous functions of $Q$, it holds that for $Q$ small enough, $\text{ker}((\bar{L}\otimes I)+\bar{Q})\subset\text{ker}(\bar{L}\otimes I)$. We assume without loss of generality that $Q$ has been chosen small enough so that the latter inclusion holds. Then (58) holds for the $R_{i}$-matrices. According to Proposition 15, if the $R_{ij}$-matrices are transitively consistent and $\mathcal{G}$ is QSC, (58) is a condition to guarantee (10). But (10) is not fulfilled since $R_{k}R_{kl}R_{l}^{-1}=(I+a_{kl}Q)\neq I$. Thus, the $R_{ij}$-matrices are not transitively consistent. $\blacksquare$

Proposition 19**.**

For any QSC-graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and collection $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ of matrices in $O(d)$ it holds that $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent if and only if there is a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $O(d)$ such that

[TABLE]

*Proof: *

Only if: Suppose that the conditions in Proposition 19 on the graph and the matrices hold. Suppose $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent. There is a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $O(d)$ such that (57) holds for the $R_{i}$-matrices. According to Proposition 15, the condition (59) also holds for the $R_{i}$-matrices.

If: Suppose $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ is not transitively consistent and that there is a collection $\{R_{i}\}_{i\in\mathcal{V}}$ of matrices in $O(d)$ such that (68) holds. Let $R_{i}(R_{ij})R_{j}^{T}=(I+Z_{ij})$ for all $(i,j)\in\mathcal{E}$. We know that there is a least one pair $(i_{1},j_{1})\in\mathcal{E}$ such that $Z_{i_{1}j_{1}}\neq 0$ (otherwise $\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}$ would be transitively consistent).

Now,

[TABLE]

where $\bar{Z}\in\mathbb{R}^{nd\times nd}$ is a block matrix whose $(i,j)$-th block is equal to $a_{ij}Z_{ij}\in\mathbb{R}^{d\times d}$.

It holds that $L_{\text{dir}}U_{1}(\{R_{i}\}_{i\in\mathcal{V}})$ is equal to zero if and only if $\bar{Z}[I,I,\ldots,I]^{T}$ is equal to zero. Thus, it holds that

[TABLE]

But it also holds, since the $(I+Z_{ij})$-matrices are orthogonal, that

[TABLE]

By combining (70) and (71) we obtain that

[TABLE]

which implies (since all the $a_{ij}>0$) that $Z_{i_{1}j_{1}}=0$. But this is a contradiction to the statement that $Z_{i_{1}j_{1}}\neq 0$. $\blacksquare$

The difference between the two previous proofs is the following. In the proof of Proposition 18 we change $L_{\text{dir}}$ with additional matrices such that the nullspace is preserved. In the proof of Proposition 19, when we try to preserve the nullspace of $L_{\text{dir}}$ under the constraint that the changed matrices are orthogonal, we reach the conclusion that the change of the matrices must be equal to zero. This means that we have to go outside the set of orthogonal matrices when we change $L_{\text{dir}}$.

5.3.2 Convergence analysis

Now we introduce the following result, in analogy to Proposition 12, but defined for asymmetric graphs.

Proposition 20**.**

Suppose convergence conditions (1) and (3-5) hold. Then, for Algorithm 2, there is a positive integer $K$ such that $\tilde{R}_{j}^{-1}(k)$ is well-defined for $k\geq K$, and (for $k\geq K$) it holds that

[TABLE]

and

[TABLE]

*Proof: * When conditions (3) and (4) hold it is easy to show that $L_{\text{dir}}$ is similar to $\bar{L}\otimes I_{d}$, where $\bar{L}$ is defined in condition (5). Since conditions (1) and (3) hold, we can use Proposition 15. There is a collection $\{\bar{R}_{i}\}_{i\in\mathcal{V}}$ such that

[TABLE]

The rest of the proof is equivalent to that of Proposition 12 and omitted. $\blacksquare$

For the general case when transitive consistency does not hold, the convergence results in this section are more conservative than those provided in Section 4.2.2 for Algorithm 1. The main difficulty is that the matrix $L_{\text{dir}}$ is not symmetric, and we cannot guarantee in general that its spectrum is real. We have the following theoretical result.

Proposition 21**.**

Suppose convergence conditions (1), (3), and (6) hold. Then

The dynamical system

[TABLE]

is critically stable. 2. 2.

$\|P\|_{2}$* is an upper bound for $\|L_{\text{dir}}\|_{2}$ where $P=\text{diag}(A1_{n})+A$. * 3. 3.

*If also the convergence condition (2) holds, *

1. (a)

*the matrix $L_{\text{dir}}$ has no purely imaginary eigenvalues, * 2. (b)

there is $\epsilon(n,d,\delta_{1},\delta_{2})>0$, such that for $\epsilon_{3}\in(0,\epsilon)$ the eigenvalues of the matrix $(I-\epsilon_{3}L_{\text{dir}}(\mathcal{G},\{{R}_{ij}\}_{(i,j)\in\mathcal{E}}))$ are contained in the closed unit disc and if there is an eigenvalue on the boundary of the disc, it is equal to $1$ and the algebraic respective geometric multiplicities thereof are equal.

*Proof: *

(1) Let

[TABLE]

where $x(0)$ is the initial state. We can write $x(t)$ as $x(t)=[x_{1}^{T}(t),x_{2}^{T}(t),\ldots,x_{n}^{T}(t)]^{T}$, where $x_{i}(t)\in\mathbb{R}^{d}$ for all $i$. Define the function $V(x)=\max_{i}(x_{i}^{T}x_{i}).$ If there is some eigenvalue of ${L_{\text{dir}}}$ with negative real part or if there is a Jordan block of dimension larger than one corresponding to an eigenvalue pair on the imaginary axis, there is $x_{0}$ such that for the state $x(t)$ with initial state $x_{0}$, $V(x(t))\rightarrow\infty$ as $t\rightarrow\infty$. We want to show that this is not possible. Let us first define the set $\mathcal{I}_{\max}(t)=\{i:V(x(t))=x_{i}^{T}(t)x_{i}(t)\}.$ Now,

[TABLE]

where $D^{+}$ is the upper Dini-derivative. A proof of the first equality in (73) can be found in [36] using the results in [37] and [38]. Now we can use the Comparison Lemma [39] to show that $V(x(t))$ is decreasing independently of the choice of $x_{0}$. The last inequality in (73) is a consequence of the fact that the ${R}_{ij}$-matrices are orthogonal.

(2) Let $v=[v_{1}^{T},v_{2}^{T},\ldots,v_{n}^{T}]^{T}\in\mathbb{S}^{nd}$, where each $v_{i}\in\mathbb{R}^{d}$. By using the structure of the $f_{2}$-function, one can show that $v^{T}L_{\text{dir}}^{T}L_{\text{dir}}v=\sum_{i=1}^{n}\|\sum_{j\in\mathcal{N}_{i}}{a_{ij}}(R_{ij}v_{j}-v_{i})\|_{F}^{2}.$ Now,

[TABLE]

(3) Now we show, by a contradiction argument, that there are no purely imaginary eigenvalues. Suppose there are purely imaginary eigenvalues. Then there must be a nontrivial periodic solution $\bar{x}(t)=[\bar{x}_{1}^{T}(t),\bar{x}_{1}^{T}(t),\ldots,\bar{x}_{n}^{T}(t)]^{T}$ to (72), i.e., $\bar{x}(t)$ is periodic and $\bar{x}(t)\neq\bar{x}(t+\Delta)$ for all $t$ and some $\Delta>0$.

It can be shown that in this particular situation $D^{+}(V(\bar{x}(t)))=0$ for all $t$ and it can also be shown that a necessary condition for this to hold is that $\bar{x}_{i}(t)=\bar{x}_{j}(t)$ for all $i,j$ and $t$, and furthermore, this can only happen if $R_{ij}=I$ for all $(i,j)$.

In the following three paragraphs we prove that the claims in the preceding paragraph are true. If $D^{+}(V(\bar{x}(t_{1})))\neq 0$, it holds that (see the proof of part (1)) $D^{+}(V(\bar{x}(t_{1})))<0$. But then $V(\bar{x}(t))<V(\bar{x}(t_{1}))$ for all $t>t_{1}$ (see the proof of part (1)), which implies that there is no $t_{2}>t_{1}$ such that $\bar{x}(t_{2})=\bar{x}(t_{2})$. Thus the solution is not periodic, which is a contradiction. Hence, we can conclude that $D^{+}(V(\bar{x}(t)))=0$ for all $t$.

Now, let us assume that there is $i_{1},j_{1}\in\mathcal{V}$ such that $\bar{x}_{i_{1}}(t_{1})\neq\bar{x}_{j_{1}}(t_{1})$ at some time $t_{1}$. Let $V_{\max}(t_{1},t)=\{k:\|x_{k}(t)\|=\max_{l\in\mathcal{V}}\|x_{l}(t_{1})\|\}.$ We can without loss of generality assume that $i_{1}\in V_{\max}(t_{1},t_{1})$, $j_{1}\not\in V_{\max}(t_{1},t_{1})$, and that $j_{1}\in\mathcal{N}_{i_{1}}$. The reason for why we can make this assumption about $i_{1}$ and $j_{1}$, is that the graph $\mathcal{G}$ is strongly connected. Now it holds that

[TABLE]

The strict inequality holds because $x_{i_{1}}^{T}(t_{1})({R}_{{i_{1}}j_{1}}x_{j_{1}}(t_{1})-x_{i_{1}}(t_{1}))<0$. We also know that $\bar{x}(t)$ is continuous and continuously differentiable. There must be $t_{2}>t_{1}$ such that $V_{\max}(t_{1},t_{2})\subset V_{\max}(t_{1},t_{1})-\{i_{1}\}.$ Now, assume that $V_{\max}(t_{1},t_{2})$ is nonempty. Then we can perform the analogous procedure again, where we pick $i_{2}\in V_{\max}(t_{1},t_{2})$, $j_{2}\not\in V_{\max}(t_{1},t_{2})$, where $j_{2}\in\mathcal{N}_{i_{2}}$, and show that that there is a $t_{3}>t_{2}$ such that $V_{\max}(t_{1},t_{3})\subset V_{\max}(t_{1},t_{2})-\{i_{2},i_{1}\}.$ By repeating this procedure, one can show by induction that there must be a $t_{f}>t_{1}$ such that $V_{\max}(t_{1},t_{f})=\emptyset$. But $V(\bar{x}(t_{f}))<V(\bar{x}(t_{1}))$ contradicts the fact that $\bar{x}(t_{f})=\bar{x}(t_{1})$.

To show that $\bar{x}_{i}(t)=\bar{x}_{j}(t)$ for all $i,j$ and $t$ can only happen when $R_{ij}=I$ for all $(i,j)$, we note the following. Suppose it holds that $\bar{x}_{i}(t)=\bar{x}_{j}(t)$ for all $i,j$ and $t$ and in particular at the time $t_{1}$. Suppose there is $i$ and an $j\in\mathcal{N}_{i}$ such that $R_{ij}\neq I_{d}$. But then one can show that the time derivative of $\|\bar{x}_{i}(t_{1})\|^{2}$ is negative at the time $t_{1}$. To see this, consider the structure of the right-hand side of (75); it holds that $\bar{x}_{i}^{T}R_{ij}\bar{x}_{j}-\bar{x}_{i}^{T}\bar{x}_{i}<0$. Thus there is a time $t_{2}>t_{1}$ where $\|\bar{x}_{i}(t_{2})\|^{2}<\|\bar{x}_{i}(t_{1})\|^{2}$. On the other hand, since $D^{+}(V(\bar{x}(t)))=0$ for all $t$, there must be a $k\in\mathcal{V}$ such that $\|\bar{x}_{k}(t_{2})\|^{2}=\|\bar{x}_{i}(t_{1})\|^{2}$. But then $\|\bar{x}_{k}(t_{2})\|^{2}\neq\|\bar{x}_{i}(t_{2})\|^{2}$, which contradicts the fact that $\bar{x}_{i}(t_{2})=\bar{x}_{k}(t_{2})$ holds.

Now, if the $R_{ij}\neq I$, the necessary condition is not fulfilled, hence we have a contradiction. In the case when the $R_{ij}=I$, it holds that $L_{\text{dir}}(\mathcal{G})=\bar{L}\otimes I_{d}$ and the latter matrix does not have purely imaginary eigenvalues.

The $R_{ij}$-matrices and the $a_{ij}$-weights are contained in a compact set parameterized by $(n,d,\delta_{1},\delta_{2})$, see convergence condtion (6). The eigenvalues of $L_{\text{dir}}$ are continuous over this set. As setforth above, there are no purely imaginary eigenvalues. Hence, any complex eigenvalue must have an argument in the open set $(\pi/2,3\pi/2)$. Supremums and infimums of continuous functions over compact sets amount to maximums and minimums, respectively, see Weierstrass’ Extreme Value Theorem. This means that the arguments of the complex eigenvalues are contained in a set $[\pi/2+\tilde{\delta}_{1},3\pi/2-\tilde{\delta}_{2}]$ for some $\tilde{\delta}_{1},\tilde{\delta}_{2}>0$. Furthermore, with analogous reasoning, the magnitudes of the eigenvalues are bounded from above by some $\tilde{\delta}_{3}$ (the magnitudes are continuous functions). Now we can construct $\epsilon$. Let $\alpha_{1}$ and $\alpha_{2}$ be the unique positive solutions to

[TABLE]

respectively. We choose $\epsilon=\min\{\alpha_{1},\alpha_{2}\}/\tilde{\delta}_{3}$.

Now, any eigenvalue that is equal to $1$ for $(I-\epsilon_{3}L_{\text{dir}})$ is corresponding to an eigenvalue that is equal to [math] for $L_{\text{dir}}$. The eigenvalues have been shifted by $1$ between the two. The eigenvectors are the same. If the claim in (3) about the multiplicity would be false, then $L_{\text{dir}}$ would be unstable, which is a contradiction, see (1). $\blacksquare$

Suppose the graph $\mathcal{G}$ is QSC and the collection of matrices $\{R_{ij}\}_{(i,j)\in\mathcal{E}}$ is transitively consistent. From Proposition 21 we can draw the following conclusion about Algorithm 2. If $\epsilon_{3}$ is chosen small enough, the system defined in (56) is stable. However, we do not provide any explicit bounds for $\epsilon_{3}$. An interesting research direction to pursue would be to investigate if the results in [40] can be used to derive such bounds.

5.4 Numerical simulations

In Figure 2 the evolution of

[TABLE]

is shown. The definition of the gap function and the problem setup are the same as in Section 4.3, apart from the construction of the graphs. The graphs are now QSC and in general asymmetric. Moreover, $\epsilon_{3}=1/(2n)$. For the top two figures the graph density is $0.9$ and for the bottom two figures the graph density is $0.5$.

There are two important things to note here. None of the results in the previous section guarantee that the algorithm converges when the graph is QSC and not necessarily strongly connected, nor do they guarantee that the algorithm converges when the step size is $1/(2n)$. Yet this was the case in all simulations we conducted. We let the graph density vary between $0.3$ and $0.9$, the dimension vary between $3$ and $20$, we tried different noise levels and $a_{ij}$-weights, and convergence to at least $-0.5$ for the gap (in logarithmic scale) was observed for the means.

6 Conclusion

In this work we proposed two distributed methods to synchronize orthogonal matrices of arbitrary dimension over graphs. One was presented for symmetric graphs and one was presented for graphs that could also be asymmetric. For both methods we provided a theoretical analysis including various convergence conditions. In the case of symmetric graphs, we proved convergence to the solution of a spectral relaxation. Numerical simulations verify this result. In the case of directed and asymmetric graphs, we proved convergence of the algorithm for small step sizes. This convergence was illustrated in numerical simulations.