CLEAR: A Consistent Lifting, Embedding, and Alignment Rectification Algorithm for Multi-View Data Association

TL;DR

CLEAR is a novel algorithm that ensures cycle consistency and improves the accuracy and efficiency of multi-view data association in robotics, enabling real-time environment modeling by addressing computational challenges and inconsistency issues.

Contribution

The paper introduces the CLEAR algorithm, which guarantees cycle consistency and computational efficiency in multi-view data association, outperforming existing methods.

Findings

Demonstrates superior accuracy on synthetic and real datasets.

Achieves real-time performance in multi-view data association tasks.

Provides a scalable framework for consistent multi-view alignment.

Abstract

Many robotics applications require alignment and fusion of observations obtained at multiple views to form a global model of the environment. Multi-way data association methods provide a mechanism to improve alignment accuracy of pairwise associations and ensure their consistency. However, existing methods that solve this computationally challenging problem are often too slow for real-time applications. Furthermore, some of the existing techniques can violate the cycle consistency principle, thus drastically reducing the fusion accuracy. This work presents the CLEAR (Consistent Lifting, Embedding, and Alignment Rectification) algorithm to address these issues. By leveraging insights from the multi-way matching and spectral graph clustering literature, CLEAR provides cycle consistent and accurate solutions in a computationally efficient manner. Numerical experiments on both synthetic and real datasets are carried out to demonstrate the scalability and superior performance of our algorithm in real-world problems. This algorithmic framework can provide significant improvement in the accuracy and efficiency of existing discrete assignment problems, which traditionally use pairwise (but potentially inconsistent) correspondences. An implementation of CLEAR is made publicly available online.

Tables

Table 1. TABLE I : Summary of important nomenclature used throughout the paper.

Notation	Domain	Definition and properties
$n$	$\mathbb{N}$	Total number of views
$m$	$\mathbb{N}$	Size of universe; number of unique items; number of cliques in the association graph
$m_i$	$\mathbb{N}$	Number of items observed in view $i$
$l$	$\mathbb{N}$	Total number of items observed across all views; $l\stackrel{\text{def}}{=}{\sum }_i{m}_i$
~	-	Accent used for variables corresponding to the noisy input
$P_j^i$	${\{0,1\}}^{m_i\times m_j}$	Partial permutation matrix; association matrix between items at views $i$ and $j$
$P$	${\{0,1\}}^{l\times l}$	Aggregate association matrix consisting of $P_j^i$’s; see (1)
$A$	${\{0,1\}}^{l\times l}$	Adjacency matrix of the association graph; $A=P-I$
$D$	${\mathbb{N}}_0^{l\times l}$	Degree matrix of the association graph
$C$	${\mathbb{N}}_0^{l\times l}$	Diagonal matrix with entries $c_i\stackrel{\text{def}}{=}{\sum }_j{\left(P\right)}_{ij}$; $C=D+I$
$L$	${\mathbb{Z}}^{l\times l}$	Laplacian matrix of $𝒢$; $L\stackrel{\text{def}}{=}D-A=C-P$
$L_{\text{nrm}}$	${ℝ}^{l\times l}$	Normalized Laplacian matrix; $L_{\text{nrm}}\stackrel{\text{def}}{=}{C}^{-\frac{1}{2}}L{C}^{-\frac{1}{2}}$
$P_{\text{nrm}}$	${ℝ}^{l\times l}$	Normalized association matrix; $P_{\text{nrm}}\stackrel{\text{def}}{=}{C}^{-\frac{1}{2}}P{C}^{-\frac{1}{2}}$
$P^i$	${\{0,1\}}^{m_i\times m}$	Lifting permutation matrix; association of items observed at views $i$ to items of the universe
$V$	${\{0,1\}}^{l\times m}$	Aggregate lifting permutation matrix consisting of $P^i$’s; see (3)
$U$	${ℝ}^{l\times m}$	Normalized aggregate lifting permutation; $U\stackrel{\text{def}}{=}{C}^{-\frac{1}{2}}V$; eigenvectors associated to $m$ smallest eigenvalues of $L_{\text{nrm}}$
$u_i$	${ℝ}^m$	Row of $U$
$u_i^{\prime }$	${ℝ}^m$	Pivot row of $U$

Table 2. TABLE II : Cross comparison of algorithms in terms of the number of definite nagatives (DN), potential positives (PP), upper bound on precision ( P ¯ ¯ P \bar{\text{P}} ), and runtime. The upper bound on precision is computed from ( 29 ).

Algorithm	DN	PP	$\overline{\text{P}}$ (%)	Runtime (s)
CLEAR	11	3393	99.677	0.084
MatchLift [10]	5	2394	99.792	124.7
MatchALS [2] (completed)	89	15230	99.419	4.580
QuickMatch [3]	897	15757	94.614	0.118
NMFSync [16]	290	3233	91.768	4.272
MatchEIG [15] (completed)	21415	20381	48.763	1.808
Baseline	26249	20487	43.836	N/A

Equations

$$P\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}\left[\begin{array}{cccc}I& P_2^1& \cdots & P_n^1\\ P_1^2& I& \cdots & P_n^2\\ \text{vbox}\text{vbox}to6.1833pt\text{offinterlineskip}\text{set@color}.\text{vfill}\text{set@color}.\text{vfill}\text{set@color}.& \text{vbox}\text{vbox}to6.1833pt\text{offinterlineskip}\text{set@color}.\text{vfill}\text{set@color}.\text{vfill}\text{set@color}.& \text{vbox}\text{vbox}to6.1833pt\text{offinterlineskip}.\text{vfill}.\text{vfill}.& \text{vbox}\text{vbox}to6.1833pt\text{offinterlineskip}\text{set@color}.\text{vfill}\text{set@color}.\text{vfill}\text{set@color}.\\ P_1^n& P_2^n& \cdots & I\end{array}\right]\in {\mathbb{R}}^{l\times l},$$

$$P_j^i=P^i{P}^{j\top },\forall i,j\in {\mathbb{N}}_n.$$

$$V\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}{\left[\begin{array}{cccc}{P}^{1\top }& P^{2\top }& \cdots & P^{n\top }\end{array}\right]}^{\top }\in \{0,1{\}}^{l\times m},$$

$$\displaystyle\begin{array}[]{ll}\underset{P^{i}_{j}\in\mathbb{P}}{\text{maximize}}&~{}\langle P_{\text{nrm}},\,\tilde{P}_{\text{nrm}}\rangle\\ \text{subject to}&~{}P=V\,V^{\top},\end{array}$$

$$\displaystyle\begin{array}[]{ll}\underset{A}{\text{maximize}}&~{}\langle A_{\text{nrm}},\,\tilde{A}_{\text{nrm}}\rangle\\ \text{subject to}&\mathcal{G}\text{~{}consists of clusters~{}}\mathcal{A}_{1},\dots,\mathcal{A}_{m}\\ &g(v_{i})\neq g(v_{j}),\quad\forall v_{i},v_{j}\in\mathcal{A}_{k}\\ \end{array}$$

$${\tilde{L}}_{\text{nrm}}\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}{\tilde{C}}^{-\frac{1}{2}}\tilde{L}{\tilde{C}}^{-\frac{1}{2}},$$

$$\hat{m}\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}\max \{\tilde{m},m_1,m_2,\dots ,m_n\},$$

$$\tilde{m}\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}\left|\{\lambda \in \mathrm{eig}({\tilde{L}}_{\text{nrm}}):\lambda <0.5\}\right|,$$

$$\tilde{P}=\left[\begin{array}{ccccccc}1& 0& 1& 0& 0& 0& 0\\ 0& 1& 0& 1& 1& 1& 1\\ 1& 0& 1& 1& 0& 0& 0\\ 0& 1& 1& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 1& 1\end{array}\right].$$

$$U=\left[\begin{array}{ccc}\text{mbox}-0.94& & 0.34\\ 0.47& & 0.88\\ \text{mbox}-0.88& & 0.48\\ 0.07& & 0.99\\ 0.47& & 0.88\\ 0.47& & 0.88\\ 0.47& & 0.88\end{array}\right].$$

$$F^1=\left[\begin{array}{cc}0& 2.28\\ 2.28& 0\end{array}\right],F^2=\left[\begin{array}{cc}0.02& 1.97\end{array}\right],F^3=\left[\begin{array}{cc}1.46& 0.17\end{array}\right],$$

$$F^4=\left[\begin{array}{cc}2.28& 0\end{array}\right],F^5=\left[\begin{array}{cc}2.28& 0\end{array}\right],F^6=\left[\begin{array}{cc}2.28& 0\end{array}\right].$$

$$P^1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],P^2=\left[\begin{array}{cc}1& 0\end{array}\right],P^3=\left[\begin{array}{cc}0& 1\end{array}\right],$$

$$P^4=\left[\begin{array}{cc}0& 1\end{array}\right],P^5=\left[\begin{array}{cc}0& 1\end{array}\right],P^6=\left[\begin{array}{cc}0& 1\end{array}\right].$$

$$\displaystyle\begin{array}[]{ll}\underset{U\in\mathbb{U}}{\text{minimize}}&\mathrm{tr}\big{(}U^{\top}\tilde{L}_{\text{nrm}}\,U\big{)},\\ \end{array}$$

$$\displaystyle\begin{array}[]{ll}\underset{U\in\mathbb{R}^{l\times m}}{\text{minimize}}&\mathrm{tr}\big{(}U^{\top}\tilde{L}_{\text{nrm}}\,U\big{)}\\ \text{subject to}&~{}U^{\top}\,U=I,\end{array}$$

$$\displaystyle\begin{array}[]{ll}\underset{\mathcal{S}\subset\mathbb{N}_{l}}{\text{minimize}}&\sum_{i,j\in\mathcal{S}}{|\langle u_{i},u_{j}\rangle|}\\ \text{subject to}&|\mathcal{S}|=m.\end{array}$$

$$\displaystyle\begin{array}[]{ll}\underset{\mathcal{A}_{1},\dots,\mathcal{A}_{m}}{\text{minimize}}&\sum_{k=1}^{m}\sum_{v_{j}\in\mathcal{A}_{k}}{\|u_{j}-u^{\prime}_{k}\|^{2}}\\ \text{subject to}&g(v_{i})\neq g(v_{j}),\quad\forall v_{i},v_{j}\in\mathcal{A}_{s}.\end{array}$$

$$F={\left[\begin{array}{cccc}{F}^{1\top }& F^{2\top }& \cdots & F^{n\top }\end{array}\right]}^{\top },$$

$$\underset{V\in \mathbb{V}}{\min }⟨V,F⟩$$

$$={\sum }_{i=1}^n\underset{P^i\in {\mathbb{P}}^L}{\min }⟨P^i,F^i⟩,$$

$$\bar{\text{P}}\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}\frac{\text{PP}}{\text{DN}+\text{PP}}.$$

$$\underset{P=V{V}^{\top }}{\max }⟨P_{\text{nrm}},{\tilde{P}}_{\text{nrm}}⟩$$

$$=\underset{V\in \mathbb{V}}{\max }\mathrm{tr}(C^{-\frac{1}{2}}V{V}^{\top }{C}^{-\frac{1}{2}}{\tilde{P}}_{\text{nrm}})$$

$$=\underset{V\in \mathbb{V}}{\max }\mathrm{tr}(V^{\top }{C}^{-\frac{1}{2}}{\tilde{P}}_{\text{nrm}}{C}^{-\frac{1}{2}}V).$$

$$\underset{V\in \mathbb{V}}{\max }$$

$$=\underset{V\in \mathbb{V}}{\max }\mathrm{tr}(V^{\top }{C}^{-\frac{1}{2}}{\tilde{P}}_{\text{nrm}}{C}^{-\frac{1}{2}}V)-\mathrm{tr}(V^{\top }{C}^{-\frac{1}{2}}{C}^{-\frac{1}{2}}V)$$

$$=\underset{U\in \mathbb{U}}{\max }\mathrm{tr}(U^{\top }{\tilde{P}}_{\text{nrm}}U)-\mathrm{tr}(U^{\top }U)$$

$$=\underset{U\in \mathbb{U}}{\max }\mathrm{tr}(U^{\top }{\tilde{C}}^{-\frac{1}{2}}\tilde{P}{\tilde{C}}^{-\frac{1}{2}}U)-\mathrm{tr}(U^{\top }{\tilde{C}}^{-\frac{1}{2}}\tilde{C}{\tilde{C}}^{-\frac{1}{2}}U)$$

$$=\underset{U\in \mathbb{U}}{\max }\mathrm{tr}(U^{\top }{\tilde{C}}^{-\frac{1}{2}}(\tilde{P}-\tilde{C}){\tilde{C}}^{-\frac{1}{2}}U)$$

$$=\underset{U\in \mathbb{U}}{\min }\mathrm{tr}(U^{\top }{\tilde{C}}^{-\frac{1}{2}}\tilde{L}{\tilde{C}}^{-\frac{1}{2}}U)$$

$$=\underset{U\in \mathbb{U}}{\min }\mathrm{tr}(U^{\top }{\tilde{L}}_{\text{nrm}}U).$$

$$\Vert {\tilde{L}}_{\text{nrm}}-L_{\text{nrm}}\Vert$$

$$=\Vert C^{-\frac{1}{2}}(\tilde{L}-L)C^{-\frac{1}{2}}\Vert$$

$$\le \Vert C^{-1}\Vert \Vert \tilde{L}-L\Vert$$

$$=\Vert C^{-1}\Vert \Vert (\tilde{D}-\tilde{A})-(D-A)\Vert$$

$$=\Vert C^{-1}\Vert \Vert \tilde{A}-A\Vert$$

$$=\Vert C^{-1}\Vert \Vert E\Vert$$

Full text

Supplementary Material

CLEAR source code and the code for generating comparison results: https://github.com/mit-acl/clear. Video of paper summary: https://youtu.be/RBxq9KYcgTY.

I Introduction

Data association across multiple views, known as the multi-view or multi-way [1] matching, is a fundamental problem in robotic perception and computer vision. Conceptually, the goal in this problem is to establish correct associations between the sightings of “items” across multiple “views” (see Fig. 1). Examples include feature matching across multiple frames [1, 2, 3], and associating landmarks across multiple maps for map fusion in single/multi-robot simultaneous localization and mapping (SLAM) [4].

The traditional approach treats the multi-view data association problem as a sequence of decoupled pairwise matching subproblems, each of which can be formulated and solved, e.g., as a linear assignment problem [5]. Such techniques, however, cannot leverage the redundancy in the observations and, furthermore, often result in non-transitive (a.k.a., cycle inconsistent) associations; see Fig. 1. One can address these issues by synchronizing all noisy pairwise associations via enforcing the cycle consistency constraint. Cycle consistency serves two crucial purposes: 1) it provides a natural mechanism for the discovery and correction of wrong (or missing) associations obtained through pairwise matching; and 2) it establishes an equivalence relation on the set of observations, which is necessary for global fusion in the so-called clique-centric applications such as map merging (Section VII).

Synchronizing pairwise associations is a combinatorial optimization problem with an exponentially large search space. This problem has been extensively studied in recent years (see [1, 2, 6, 3] and references therein). These efforts have resulted in several algorithms that can improve the erroneous initial set of pairwise associations. However, providing solutions that are computationally tractable for real-time applications remains a fundamental challenge. Further, the rounding techniques used by some of relaxation-based methods may violate the cycle consistency and distinctness constraints (distinctness implies that the items observed in each view are unique, and thus must not be associated with each other).

This work aims to address these critical challenges via a novel spectral graph-theoretic approach. Specifically, we leverage the natural graphical representation of the problem and propose a spectral graph clustering technique uniquely tailored for producing accurate solutions to the multi-way data association problem in a computationally tractable manner. Our solutions, by construction, are guaranteed to satisfy the cycle consistency and distinctness constraints under any noise regime. These are demonstrated in our extensive empirical evaluations based on synthetic and real datasets in the context of feature matching and map fusion in landmark-based SLAM.

I-A Related Work

With the exception of combinatorial methods [7, 8] that do not scale well to large problems, and a recent deep leaning approach in [9], the majority of permutation synchronization algorithms that aim to solve this computationally challenging problem can be classified as (i) convex relaxation; (ii) spectral relaxation; and (iii) graph clustering.

Methods in the first category include [10], which uses a semidefinite programming relaxation of the problem and solves it via ADMM [11]. A distributed variation of this method with a similar formulation has been recently presented in [12]. Toward the same goal, [2] uses a low-rank matrix factorization to improve the computational complexity. Works such as [13] and [6] require full observability, whereas methods such as [14] can perform in a partially observable setting, where only a subset of overall items is observed at each view. The aforementioned algorithms often return solutions that have the highest accuracy; however, due lifting to high dimensional spaces, they are slow and not suitable for real-time applications.

Methods in the second category are based on a spectral relaxation of the problem, with prominent works including [1] and [15]. The method proposed in [1] returns consistent solutions from noisy pairwise associations using a spectral relaxation in the fully observable setting. The work done by [15] proposes an eigendecomposition approach that works in a partially observable setting, however cycle consistency is lost in higher noise regimes. The recent work of [16] leverages a non-negative matrix factorization approach to solve the problem. This method works in a partially observable setting and preserves cycle consistency. Algorithms that use spectral relaxation are relatively fast and return solutions that have comparably high accuracy.

Methods in the third category use a graph representation of the problem. In [17] and [3], the authors have elegantly observed the equivalence relation between cycle consistency and cluster structure of the association graph. This observation is used to find approximate solutions to the problem based on existing graph clustering algorithms. The work done in [17] has considered a constrained clustering approach using a method similar to $k$-means. In [3], the existing density-based graph clustering algorithm in [18] is leveraged to solve the problem. Methods in this category could be very fast, however, the accuracy may be compromised.

Lastly, we point out that the multi-way data association problem can be viewed and solved from a graph matching perspective [19, 20]. Unlike all previously discussed methods (and the present work), which only leverage the association information across views, graph matching additionally incorporates geometrical information between the items in each view. The additional complexity, in general, results in significantly slower algorithms.

I-B Our Contributions

Our work provides new insights into connections between the multi-way data association problem and the spectral graph clustering literature. We leverage these insights to push the boundaries of accuracy and speed—which are crucial for real-time robotics applications—to solve the multi-way data association problem. The main contributions of this work are as follows:

To our knowledge, this work provides the first approach that formulates and solves the multi-way association problem using a normalized objective function. This normalization is crucial to recover the correct solution when the association graph is a mixture of large and small clusters (Remark 1). 2. 2.

We leverage the natural graphical structure of the problem to estimate the unknown universe size111By definition, universe size is the total number of unique items in all views. from erroneous associations. Specifically, we prove that our technique is guaranteed to recover the correct universe size under certain bounded noise regimes (Proposition 2). Moreover, we empirically demonstrate that the proposed approach is more robust to noise than the standard eigengap heuristic [21] used in the spectral graph clustering literature (Remark 3). 3. 3.

We propose a projection (rounding) method that, by construction, is guaranteed to produce solutions that satisfy the cycle consistency and distinctness constraints, whereas these constraints can be violated by some of the state-of-the-art algorithms in high-noise regimes (Section VIII).

In addition, we address an important subtlety regarding the choice of suitable metrics for evaluating the performance of multi-way matching algorithms in applications such as map fusion (Section VII). Finally, we provide extensive numerical experiments on both synthetic and real datasets in the context of feature matching and map fusion (Sections VIII and IX). Our empirical results demonstrate the superior performance of our algorithm in comparison to the state-of-the-art methods in terms of both accuracy and speed.

Outline

The organization of the paper is as follows. The notation and definitions are introduced in Section II, followed by the problem formulation in Section III. The CLEAR algorithm is presented in Section IV, followed by a numerical example in SectionV. The theoretical justifications behind the algorithm are discussed in Section VI with proofs presented in the Appendix. Application domains for the CLEAR algorithm are discussed in Section VII. CLEAR is benchmarked against the state-of-the-art algorithms using synthetic data in Section VIII.Finally, experimental evaluations of CLEAR on real-world datasets are presented in Section IX.

II Notation and Definitions

We denote the set of natural numbers by $\mathbb{N}$, integers by $\mathbb{Z}$, $\mathbb{N}_{0}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\{0\}\cup\mathbb{N}$, and define $\mathbb{N}_{n}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\{1,\,2,\,\cdots,n\}$. Scalars and vectors are denoted by lower case (e.g., $a$), matrices by uppercase (e.g., $A$), and sets by script letters (e.g., $\mathcal{A}$). Cardinality of set $\mathcal{A}$ is denoted by $|\mathcal{A}|$. The element on row $i$ and column $j$ of matrix $A$ is denoted by $(A)_{ij}$. The Frobenius inner product is defined as $\langle A,B\rangle\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\mathrm{tr}(A^{\top}B)$, where $A$ and $B$ are matrices of the same size. Finally, $\|\cdot\|$ denotes the (induced) 2-norm. Table I lists the key variables used throughout the paper.

II-A Permutation Matrices

Matrix $P^{i}_{j}\in\{0,1\}^{m_{i}\times m_{j}}$ is said to be a partial permutation matrix if and only if each row and column of $P^{i}_{j}$ at most contains a single 1 entry. Matrix $P$ is called a full permutation matrix if and only if each row and column has exactly a single 1 entry. We denote the space of all (partial or full) permutation matrices by $\mathbb{P}$. Matrix $P^{i}\in\mathbb{P}$ is said to be a lifting permutation matrix if and only if each row of $P^{i}$ contains a single 1 entry (however, column entries could be all zero). We denote the space of all lifting permutation matrices by $\mathbb{P}^{L}$. The aggregate association matrix consisting of matrices $P^{i}_{j}\in\{0,1\}^{m_{i}\times m_{j}}$, $i,j\in\mathbb{N}_{n}$, is defined as

[TABLE]

where $I$ is the identity matrix with appropriate size, and ${l\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\sum_{i=1}^{n}{m_{i}}}$.

II-B Graph Theory

We denote a graph with $l$ vertices by $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of vertices, and $\mathcal{E}$ is the set of undirected edges. The adjacency matrix $A\in\{0,1\}^{l\times l}$ of $\mathcal{G}$ is defined by $(A)_{ij}=a_{ij}$, where $a_{ij}=1$ if there is an edge between vertices ${v_{i},v_{j}\in\mathcal{V}}$, otherwise $a_{ij}=0$. We assume $a_{ii}=0$, i.e., graph has no self-loops. The degree of a vertex $v_{i}\in\mathcal{V}$ is defined as ${d_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\sum_{j=1}^{l}{a_{ij}}}$, and the $l\times l$ degree matrix $D$ is defined as a diagonal matrix with $d_{1},\dots,d_{l}$ on the diagonal. We define $C\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}D+I$, where $I$ is identity matrix. If $c_{i}$’s denote the diagonal entries of $C$, then $C^{-\frac{1}{2}}$ is a diagonal matrix with diagonal entries $1/\sqrt{c_{i}}$. The Laplacian matrix of $\mathcal{G}$ is defined as $L\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}D-A$. A cluster graph $\mathcal{G}$ is a disjoint union of cliques (i.e., complete subgraphs). That is, $\mathcal{G}$ can be partitioned into subgraphs $\mathcal{A}_{1},\mathcal{A}_{2},\dots,\mathcal{A}_{m}$, where each $\mathcal{A}_{i}$ is a complete graph and there is no edge between any two $\mathcal{A}_{i},\,\mathcal{A}_{j}$. The cliques in a cluster graph are called clusters.

III Problem Formulation

Simply put, the objective of this paper is to reconstruct a set of cycle consistent associations from a set of pairwise associations, which may contain error and lack cycle consistency. This problem can be approached from either an optimization or a graph-theoretic viewpoint. In what follows, we will first describe each formulation separately, and then shed light on their connections.

III-A Optimization-Based Formulation

We consider $n$ views and assume that view $i$ contains $m_{i}$ items. Associations (or matchings) between items in views $i$ and $j$ can be represented by a binary matrix ${P^{i}_{j}\in\{0,1\}^{m_{i}\times m_{j}}}$, in which the one entries indicate the associations. An example of pairwise associations among three views is shown in Fig. 2. A lifting permutation represents the association between items observed in a view and the universe (which by definition consists of all items). An example is provided in Fig. 3.

Definition 1**.**

Pairwise associations $P^{i}_{j}$ are cycle consistent if and only if there exist lifting permutations ${P^{i}\in\mathbb{P}^{L}}$ such that

[TABLE]

The cycle consistency condition (2) can be presented more concisely as $P=V\,V^{\top}$, where $P$ is the aggregate association matrix defined in (1), and

[TABLE]

where $l\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\sum_{i}{m_{i}}$. Here, $m\in\mathbb{N}_{l}$ is the number of columns of lifting permutations that is referred to as the size of universe.

Throughout this paper, we use the accent ~ to distinguish the variables that are associated with the noisy input. Therefore, ${\tilde{P}^{i}_{j}\in\{0,1\}^{m_{i}\times m_{j}}}$ denotes the noisy association between items in views $i$ and $j$, where $\tilde{P}^{i}_{i}=I$ by definition. Note that $\tilde{P}^{i}_{j}$’s can be erroneous and inconsistent. Let $\tilde{P}\in\mathbb{R}^{l\times l}$, defined via (1), denote the noisy aggregate association matrix. Further, let $\tilde{C}$ be an $l\times l$ diagonal matrix with positive diagonal entries $\tilde{c}_{1},\dots,\tilde{c}_{l}$ defined as the sum of corresponding rows of $\tilde{P}$, i.e., $\tilde{c}_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\sum_{j=1}^{l}{(\tilde{P})_{ij}}$. Using definitions above, we now formulate the main problem.

Problem 1**.**

Given noisy associations $\tilde{P}^{i}_{j}$, find cycle consistent associations $P^{i}_{j}$ that solve the program

[TABLE]

where $P_{\text{nrm}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}C^{-\frac{1}{2}}\,P\,C^{-\frac{1}{2}}$, $\tilde{P}_{\text{nrm}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\tilde{C}^{-\frac{1}{2}}\,\tilde{P}\,\tilde{C}^{-\frac{1}{2}}$.

In Problem 1, diagonal matrices $C$ and $\tilde{C}$ are used to normalize the aggregate association matrices. The justification behind this normalization will be explained in Remark 1 after the graph formulation of the problem is introduced. The constraint $P^{i}_{j}\in\mathbb{P}$ enforces the permutation structure, preventing the rows and columns of $P^{i}_{j}$ from having more than a single one entry. This enforces the distinctness constraint, which implies that items in the same view are unique, thus should not be associated with each other. The constraint $P=V\,V^{\top}$ imposes cycle consistency, capturing the fact that correct associations should be transitive (i.e., if item $i$ is associated to item $j$, and item $j$ is associated to item $k$, then item $i$ must also be associated to item $k$).

III-B Graph-Based Formulation

The problem of data association has a graph representation. This representation provides the key insights that are leveraged by our algorithm to improve accuracy and runtime. A set of pairwise associations $P^{i}_{j}$ can be represented as a colored graph, where items in each view are denoted by vertices with identical color, and each nonzero entry of $P^{i}_{j}$ represents an edge between the corresponding vertices (e.g., Fig. 2). Formally, an association graph is defined as $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with the coloring map $g:\mathcal{V}\rightarrow\mathbb{N}_{n}$. The set of vertices $\mathcal{V}$ consists of subsets $\mathcal{V}_{i}$ corresponding to items in view $i$, where $g(v_{j})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}i$ for all $v_{j}\in\mathcal{V}_{i}$. The set of edges $\mathcal{E}$ consists of subsets $\mathcal{E}_{ij}$, $i\neq j\in\mathbb{N}_{n}$, corresponding to associations, where $\{v_{r},v_{s}\}\in\mathcal{E}_{ij}$ if and only if $(P^{i}_{j})_{rs}=1$.

The variables $P,\,C$ and $V$ defined previously in the optimization formulation (6) also have graph interpretations. Specifically, the adjacency matrix of the association graph $\mathcal{G}$ is given by $A=P-I$. Further, we have that $C=D+I$, where $D$ is the degree matrix of the graph. To understand the graph interpretation of $V$, we first note the following relation between the cycle consistency and the graph representation.

Proposition 1**.**

A set of pairwise associations is cycle consistent if and only if the corresponding association graph is a cluster graph (i.e., a disjoint union of complete subgraphs).

Proof of Proposition 1 is given in [3, Prop. 2] and hence omitted here. The proof reveals the connection between the algebraic definition of cycle consistency, $P=V\,V^{\top}$, and clusters of the association graph, denoted by $\mathcal{A}_{1},\dots,\mathcal{A}_{m}$. In particular, row $i$ of the aggregate lifting permutation matrix $V\in\{0,1\}^{l\times m}$ represents vertex $v_{i}$ of the association graph. The one entries in $j$-th column of $V$ indicate the vertices that belong to cluster $\mathcal{A}_{j}$ of $\mathcal{G}$. That is, if $(V)_{ij}=(V)_{kj}=1$, then vertices $v_{i}$ and $v_{k}$ are connected by an edge and belong to cluster $\mathcal{A}_{j}$. We will leverage this observation in the theoretical analysis of the algorithm.

Given a noisy association graph $\tilde{\mathcal{G}}$ with adjacency matrix $\tilde{A}$, degree matrix $\tilde{D}$, and $\tilde{C}=\tilde{D}+I$, the graph-based formulation of the multi-way association problem is as follows.

Problem 2**.**

Given the noisy association graph $\tilde{\mathcal{G}}$, find undirected graph $\mathcal{G}$ with adjacency matrix $A$ that solves

[TABLE]

where $A_{\text{nrm}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}C^{-\frac{1}{2}}\,A\,C^{-\frac{1}{2}}$ and $\tilde{A}_{\text{nrm}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\tilde{C}^{-\frac{1}{2}}\,\tilde{A}\,\tilde{C}^{-\frac{1}{2}}$.

Note that Problems 1 and 2 are equivalent. As elaborated above, the indices of the vertices belonging to clusters $\mathcal{A}_{1},\,\dots,\mathcal{A}_{m}$ uniquely determine $V$ in Problem 1. Further, since $A=P-I$, both objective functions have the same optimizer. In (10), the first two constraints respectively correspond to the cycle consistency and distinctness of associations, where the latter is achieved by the fact that the colors of vertices in each cluster must be distinct.

Remark 1**.**

The normalized objective function in (6) is a key distinction from several state-of-the-art methods [1, 15, 16] that consider the unnormalized objective $\langle P,\,\tilde{P}\rangle$. By weighting edges based on the degrees of their adjacent vertices, the normalized objective provides a measure to “balance” the number of edges that are removed from or added to the noisy association graph $\tilde{\mathcal{G}}$ to obtain $\mathcal{G}$. The unnormalized objective, on the other hand, is indifferent to the number of added edges. This can lead to (undesired) optimal solutions that consist of many additional edges. This point is illustrated in Fig. 4, where, in contrast to the normalized objective, the optimal solution with an unnormalized objective could fail to recover the ground truth even in a relatively low-noise regime.

We point out that the example shown in Fig. 4 is only one of countless scenarios in which the optimal solution of an unnormalized objective could fail to recover the desired association in a low-noise regime. Such examples can be constructed by (wrongly) associating clusters with small and large number of vertices.

IV The Consistent Lifting, Embedding, and Alignment Rectification (CLEAR) Algorithm

In this section, we present a concise description of the CLEAR algorithm used for solving the permutation synchronization problem, followed by a numerical example to further illustrate the steps of the algorithm in the next section. Theoretical justifications of the algorithm will be discussed in Section VI. The pseudocode of CLEAR is given in Algorithm 1, where each step is discussed in details below.

$\bullet$ Step 1: Let $\tilde{\mathcal{G}}$ denote the association graph corresponding to a set of noisy pairwise associations $\tilde{P}^{i}_{j}$. Define the normalized Laplacian of $\tilde{\mathcal{G}}$ as

[TABLE]

where $\tilde{L}=\tilde{D}-\tilde{A}$, $\tilde{C}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\tilde{D}+I$, and $\tilde{D},\,\tilde{A}$ are respectively the degree and adjacency matrix of $\tilde{\mathcal{G}}$. Compute the eigenvalues and eigenvectors of $\tilde{L}_{\text{nrm}}$.

To reduce the computational complexity, eigendecomposition of $\tilde{L}_{\text{nrm}}$ is done by first finding the connected components of $\tilde{\mathcal{G}}$ using the breadth-first search (BFS) algorithm [22]. Eigenvalues of $\tilde{L}_{\text{nrm}}$ are then given as the disjoint union of each component’s normalized Laplacian eigenvalues. Similarly, eigenvectors are given by appropriately padding the eigenvectors of connected components with zeros.

We point out that if $\tilde{L}_{\text{nrm}}$ is not symmetric, its symmetric component $(\tilde{L}_{\text{nrm}}+\tilde{L}_{\text{nrm}}^{\top})/2$ should be used instead in the eigendecomposition (the skew-symmetric component does not contribute to the optimal answer; see Remark 2). Note that the symmetry implies that all eigenvalues and eigenvectors are real.

$\bullet$ Step 2: Obtain an estimate for the size of universe as

[TABLE]

where $m_{i}$ is the number of items in view $i$, and $\tilde{m}$ is defined as

[TABLE]

i.e., the number of eigenvalues of $\tilde{L}_{\text{nrm}}$ that are less than $0.5$.

$\bullet$ Step 3: Define matrix $U\in\mathbb{R}^{l\times m}$ as the $\hat{m}$ first eigenvectors of $\tilde{L}_{\text{nrm}}$, that is, the eigenvectors associated with the smallest eigenvalues.

$\bullet$ Step 4: Normalize rows of $U$ to have unit norm, i.e., the $i$-th row of $U$, denoted by $u_{i}$, is replaced by $u_{i}/\|u_{i}\|$. Choose the $\hat{m}$ most orthogonal rows as pivots.

This can be done based on a greedy strategy where the first row of $U$ is chosen as the first pivot. To find the remaining pivots, the row with the smallest inner product magnitude with previously chosen pivots is picked consecutively. That is, if $u^{\prime}_{k}$ denotes the $k$-th pivot, $u^{\prime}_{k+1}$ is selected such that $\sum_{i=1}^{k}{|\langle u^{\prime}_{i},u^{\prime}_{k+1}\rangle|}$ is minimized.

For each view $i$, define matrix ${F^{i}\in\mathbb{R}^{m_{i}\times m}}$ by ${(F^{i})_{jk}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\|u_{j}-u^{\prime}_{k}\|^{2}}$, where $u_{j}$ denotes the rows of $U$ associated to items in view $i$, and $u^{\prime}_{k}$ denotes the pivot rows.222Specifically, $u_{j}$ denotes rows $\sum_{k=1}^{i-1}m_{k}+1$ through $\sum_{k=1}^{i}m_{k}$ of $U$. Solve a linear assignment problem based on $F^{i}$ as the cost matrix to obtain a lifting permutation $P^{i}\in\mathbb{P}^{L}$ that associates the items in view $i$ (rows $u_{j}$) to the items of the universe (pivot rows $u^{\prime}_{k}$). The Hungarian algorithm [5] can be used to solve the linear assignment problem and find the optimal answer. However, to reduce the computational complexity, faster (suboptimal) algorithms can be used instead while the distinctness constraint is preserved by ensuring that each $u^{\prime}_{k}$ is associated to at most one $u_{j}$, and each $u_{j}$ is associated to exactly one $u^{\prime}_{k}$.

$\bullet$ Step 5: Compute pairwise associations as $P^{i}_{j}=P^{i}P^{j\,\top}$. From Definition 1, pairwise associations are cycle consistent by construction.

V Numerical Example

We present an example to illustrate the steps of the CLEAR algorithm and show how pivot rows are chosen.

Example 1**.**

In this example, we use the CLEAR algorithm to recover cycle-consistent associations from the (noisy) association graph $\tilde{\mathcal{G}}$ shown in Fig. 5. Note that $\tilde{\mathcal{G}}$ is identical to $\mathcal{G}_{1}$ in Fig. 4, where the correct associations and the labels $A,B$ are unknown and should be recovered. The aggregate association matrix (which is equal to the adjacency matrix plus identity) is given by

[TABLE]

The first two rows of $P$ correspond to items in view 1 and the remaining rows successively correspond to views 2 through 6.

$\bullet$ Step 1: From (14), the Laplacian matrix is computed as $\tilde{L}=\tilde{C}-\tilde{P}$, where $\tilde{C}=\mathrm{diag}(2,5,3,6,5,5,5)$ and $\mathrm{diag}$ creates a diagonal matrix from input arguments. The normalized Laplacian matrix is given by $\tilde{L}_{\text{nrm}}=\tilde{C}^{-\frac{1}{2}}\,\tilde{L}\,\tilde{C}^{-\frac{1}{2}}$, which has eigenvalues $\{1.18,\,1,\,1,\,1,\,0.85,\,0.17,\,0\}$.

$\bullet$ Step 2: The number of eigenvalues of $\tilde{L}_{\text{nrm}}$ that are less than $0.5$ are two. Hence, $\tilde{m}=2$. The number of items in views is either two (for view 1) or one (for the rest of views). Thus, the estimated size of universe is obtained as $\hat{m}=2$.

$\bullet$ Step 3: Matrix $U$ consisting of the first two eigenvectors of $\tilde{L}_{\text{nrm}}$ is computed.

$\bullet$ Step 4: Rows of $U$ are normalized to obtain (up to two decimals)

[TABLE]

Fig. 6 depicts rows of (15) as vectors, where the endpoint of each vector is colored based on the view that it corresponds to, and the unit circle is drawn to indicate that rows have unit norm. The pivots are chosen by taking the first row as the first pivot $u^{\prime}_{1}=[-0.94,\,0.34]$. The second pivot is chosen as the row of $U$ that has the smallest (absolute value of) inner product with $u^{\prime}_{1}$, which gives $u^{\prime}_{2}=[0.47,\,0.88]$.

From $(F^{i})_{jk}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\|u_{j}-u^{\prime}_{k}\|^{2}$, where $u_{j}$ are rows of $U$ that correspond to view $i$ and $u^{\prime}_{k}$ are pivot rows we obtain

[TABLE]

By solving a linear assignment problem for each $F^{i}$ as the cost matrix (which aims to find the permutation matrix $P^{i}$ such that $\langle P^{i},F^{i}\rangle$ is minimized) we obtain lifting permutations

[TABLE]

$\bullet$ Step 5: Cycle-consistent pairwise associations are obtained by $P^{i}_{j}=P^{i}P^{j}$. Note that these associations correspond to the graph $\mathcal{G}_{3}$ in Fig. 4.

VI Theoretical Justifications

In this section, we discuss the insights and theoretical justifications behind steps of the CLEAR algorithm. To improve the readability, proof of all lemmas and propositions are presented in the Appendix.

The discrete and combinatorial nature of the multi-way data association problem makes finding the optimal solution computationally prohibitive in practice. Hence, similar to the state-of-the-art methods, the CLEAR algorithm aims to find a suboptimal solution via a series of approximations of the original problem.

VI-A Step 1: Reformulation

Before proceeding with obtaining an approximate solution, we reformulate (6) to obtain an equivalent problem. This equivalent problem, given in the following proposition, is amenable to a relaxation, which grants us an approximate solution in a computationally tractable manner.

Proposition 2**.**

Problem 1 is equivalent to

[TABLE]

where $\mathbb{U}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\{U\,:\,U=C^{-\frac{1}{2}}V,\,V\in\mathbb{V}\}$, $\mathbb{V}$ is defined as the set of all matrices of form (3), and $U^{\top}U=I$.

Remark 2**.**

The skew-symmetric part of $\tilde{L}_{\text{nrm}}$ does not affect the solution of (17) since for all $U$ and any skew-symmetric matrix $B$, $\mathrm{tr}(U^{\top}B\,U)=0$. This observation justifies using only the symmetric part of $\tilde{L}_{\text{nrm}}$ in step 1 of the CLEAR algorithm.

VI-B Step 2: Estimating Size of Universe

From (12) and (13), CLEAR obtains an estimate for the size of universe based on the spectrum of $\tilde{L}_{\text{nrm}}$. By definition, the size of universe is the total number of unique items observed in all views (e.g., the size of universe in Fig. 3 is five), which essentially determines the number of columns of $U$ in (17) (or equivalently $V$ in (6)). This approach is justified in the following analysis, which aims to show that, under certain bounded noise regimes, the estimated size $\hat{m}$ is guaranteed to be identical to its true value $m$. Let us denote the ground truth association graph by $\mathcal{G}$. Note that $\mathcal{G}$ consists of $m$ clusters, each representing an item of the universe.

Lemma 1**.**

If $L_{\text{nrm}}$ is the normalized Laplacian matrix of the cluster graph $\mathcal{G}$, then $\mathrm{eig}(L_{\text{nrm}})$ consists of zeros and ones. Moreover, the multiplicity of the zero eigenvalues is the number of clusters.

Lemma 1 implies that in the noiseless setting the number of zero eigenvalues of $L_{\text{nrm}}$ is the size of universe, which is correctly recovered from (13) by counting the eigenvalues that are less than $0.5$. We now consider the noisy association graph $\tilde{\mathcal{G}}$ with normalized Laplacian $\tilde{L}_{\text{nrm}}=L_{\text{nrm}}+N$, where $N$ is a symmetric matrix that represents the noise. Here, the symmetry assumption follows from using only the symmetric component of $\tilde{L}_{\text{nrm}}$ in the algorithm (see Remark 2).

Lemma 2**.**

Consider the estimate $\tilde{m}$ obtained by (13) from $\tilde{L}_{\text{nrm}}=L_{\text{nrm}}+N$. If $\|N\|<0.5$, then $\tilde{m}=m$.

Lemma 2 implies that, under a bounded noise regime, the estimated size of universe is equal to the true value. In order to obtain a bound in terms of the number of wrong associations for which $\tilde{m}=m$ is guaranteed, let us consider a noise model where $\tilde{C}=C$. In this model, correct associations are potentially replaced with wrong ones, however, the degrees of vertices in $\mathcal{G}$ and $\tilde{\mathcal{G}}$ remain the same. Let $\tilde{A}=A+E$, where $A$ and $\tilde{A}$ are respectively the adjacency matrices of $\mathcal{G}$ and $\tilde{\mathcal{G}}$, and $E\in\{-1,0,1\}^{l\times l}$ represents the noise. Further, let $e_{\max}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\max{\{e_{1},\,e_{2},\,\dots,\,e_{l}\}}$, where $e_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\sum_{j=1}^{l}{|(E)_{ij}|}$ denotes the number of wrong associations at vertex $i$ of the graph $\tilde{\mathcal{G}}$. Let $c_{\min}>0$ denote the smallest diagonal entry of the $C$ matrix.

Proposition 3**.**

Given $e_{\max},\,c_{\min}$ defined above and $\tilde{m}$ obtained from (13), if $e_{\max}<0.5\,c_{\min}$, then $\tilde{m}=m$.

Proposition 3 implies that when the noise magnitude (in terms of the number of mismatches) is sufficiently small, the estimated size of universe $\hat{m}$ is equal to the true value $m$. We point out that in practice the bound in Proposition 3 is conservative and correct estimates may be obtained in larger noise regimes or for noise with a more realistic model. In higher noise regimes where the estimate can have a large error, taking the maximum in (12) ensures the distinctness constraint (which implies that items in each view are unique), and therefore the estimated $m$ cannot be less than the maximum number of items observed at a view.

The estimated value of $m$ obtained from (12) fixes the size of $U$ in (17) throughout the algorithm. Since (as we will show) each iteration of the CLEAR algorithm has a small execution time, instead of using a fixed value an alternative approach is to consider multiple values for $\hat{m}$ (e.g., by looping over all feasible $\hat{m}$) and choosing the solution that maximizes the objective in (6). In our empirical evaluations we observed that this approach, which comes at the expense of a higher execution time, does not notably improve the accuracy of the results. This empirical observation hints that the estimated value of $\hat{m}$ is often close to its optimal value, advocating the proposed estimation approach.

Remark 3**.**

In the spectral graph clustering methods, the “eigengap” heuristic is often used to estimate the number of clusters [21]. In this approach, $\tilde{m}$ is chosen such that ${|\lambda_{\tilde{m}}-\lambda_{\tilde{m}+1}|}$ is maximized, where $\lambda_{k}$’s are sorted eigenvalues of ${L_{\text{sym}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}D^{-\frac{1}{2}}LD^{-\frac{1}{2}}}$. Unlike the normalized Laplacian $L_{\text{nrm}}$ proposed in this work (see Lemma 1), in the noiseless setting, the nonzero eigenvalues of $L_{\text{sym}}$ depend on the size of clusters. We believe that this can make the eigengap method more sensitive to noise. As we will see in our empirical analysis in Section VIII, the estimated universe size based on the eigengap heuristic can deviate considerably from the true value in moderate noise regimes, while our approach exhibits more robustness.

VI-C Step 3: Lifting and Relaxation

In order to solve (17) in a computationally tractable manner, the second approximation used in the CLEAR algorithm is to drop the discrete constraint $U\in\mathbb{U}$ and allow $U$ to take values in $\mathbb{R}^{l\times m}$. This leads to the relaxed problem

[TABLE]

which is a generalized Rayleigh quotient problem. From the Rayleigh-Ritz theorem [23, Sec 5.2.2], it follows that the solution of (20) is given by the eigenvectors corresponding to the $m$-smallest eigenvalues of $\tilde{L}_{\text{nrm}}$ (note that $m$ is estimated in the previous step).

We point out that the relaxation technique used here is similar to the relaxation used to solve the normalized minimum-cut problem in the spectral graph clustering literature [21]. This similarity is not surprising given the graph-theoretic interpretation of our problem discussed in Section III-B. Nevertheless, note that spectral graph clustering is based on $\tilde{L}_{\text{sym}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\tilde{D}^{-\frac{1}{2}}\tilde{L}\tilde{D}^{-\frac{1}{2}}$ (or other normalized Laplacians) instead of $\tilde{L}_{\text{nrm}}$.

VI-D Step 4: Projection and Embedding

In order to obtain an approximate solution for the original problem (17), the solution $U^{*}\in\mathbb{R}^{l\times m}$ obtained from solving (20) should be projected back to the discrete set $\mathbb{U}$. This step is critical for ensuring the cycle consistency and distinctness constraints. In fact, as we will show in Section VIII, the solutions returned by some state-of-the-art methods could violate the cycle consistency or distinctness constraints in high-noise regimes due to bad projections.

To project $U^{*}$ onto $\mathbb{U}$, several approaches can be considered. Our approach is inspired by the spectral graph clustering literature [21, 24, 25], where rows of $U^{*}$ are normalized and embedded as points in $\mathbb{R}^{m}$. These points are then grouped into $m$ disjoint sets based on their distance to cluster centers. Despite the aforementioned similarity, a key difference in our setting is the existence of the distinctness constraint (i.e., vertex coloring), which is not present in spectral graph clustering [25]. Hence, the $k$-means algorithm commonly used for grouping the embedded points in general violates the distinctness constraint. Furthermore, compared to other projection techniques that consider this constraint (e.g., methods in [26, 16]), our approach has a lower complexity that leads to superior execution time.

Our approach is based on noting that rows of $V$ (defined in (3)) consist of standard bases vectors which are orthogonal. Furthermore, as explained earlier, $V$ identifies graph clusters $\mathcal{A}_{1},\mathcal{A}_{2},\dots,\mathcal{A}_{m}$, where vertices that belong to the same cluster have identical rows in $V$. Since $U\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}C^{-\frac{1}{2}}\,V$ and $C$ is a diagonal matrix, it follows that in the noiseless setting the set of normalized rows of $U$ consists exclusively of $m$ orthogonal vectors. Additionally, similar to $V$, normalized rows of $U$ that are identical correspond to vertices that belong to the same cluster.

In the noisy setting, from the Davis-Kahan theorem [27] the eigenspace of the ground truth Laplacian matrix and its noisy version are “close” to each other (where “closeness” can be quantified by the noise magnitude, cf. the discussion in [21, Sec. 7]). Hence, in modest noise regimes, the rows of $U^{*}$ that belong to the same clusters are expected to remain close (in terms of the Euclidean distance) and almost perpendicular to other rows. This observation is leveraged by choosing $m$ rows of $U^{*}$ that are most orthogonal to each other (called pivots) to represent the clusters. The remaining rows are then associated to pivots (while preserving distinctness) based on distance in order to identify which cluster they belong to.

If $u_{i}$ denotes the $i$-th row of $U^{*}$, the problem of finding the $m$ most orthogonal rows can be formulated as finding a subset $\mathcal{S}$ of rows that solves

[TABLE]

The greedy strategy explained in step 3 of the CLEAR algorithm can be leveraged to efficiently find an approximate solution for (23).

After choosing the pivot rows, which are denoted by $u^{\prime}_{k}$ and represent clusters, the remaining rows of $U^{*}$ are assigned to pivot rows through minimizing the within-cluster distances. This is formally stated as

[TABLE]

The constraint in (26) enforces the distinctness constraint (i.e., items in a view should not be in the same cluster). Let us define $F\in\mathbb{R}^{l\times m}$ such that $(F)_{jk}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\|u_{j}-u^{\prime}_{k}\|^{2}$, and denote its row blocks by

[TABLE]

where the number of rows of block $F^{i}$ is equal to the number of items at view $i$. Using this notation, and since $V$ encapsulates both the distinctness constraint and the cluster structure,333If the $j$-th entry in column $k$ of $V$ is nonzero, then $v_{j}\in\mathcal{A}_{k}$. (26) can be represented in matrix form as $\underset{V\in\mathbb{V}}{\mathrm{min}}~{}\langle V,\,F\rangle$. Noting that

[TABLE]

and since each subproblem in (28b) is a linear assignment problem [5], the optimal solution can be obtained by, e.g., applying the Hungarian (Kuhn-Munkres) algorithm on each block $F^{i}$.

From the implementation point of view, as long as the lifting permutation structure of $P^{i}$ is preserved, faster suboptimal methods can be used instead to solve (28b). To improve the runtime, instead of the Hungarian algorithm CLEAR uses a suboptimal greedy strategy based on sorting, where the index of the smallest entries of $F^{i}$ are used to determine the index of one entries in $P^{i}$. These indices are chosen with care to ensure that $P^{i}$ is a lifting permutation (i.e., each row has a single one entry and each column has at most a single one entry). In our empirical evaluations we observed that this suboptimal strategy performs as well as the optimal Hungarian algorithm most of the time in term of accuracy, but has a considerable speed advantage.

Lastly, we emphasize that the proposed projection technique is based on the orthogonality property of the embedded rows. Hence, the results are not affected by any transformation that preserves the orthogonality. This is particularly important since solutions of (20) are only recovered up to an orthogonal transformation (i.e., if $U^{*}$ is a solution so is $U^{*}R$ for any orthogonal matrix $R$).

VI-E Computational Complexity

The computational complexity of CLEAR is determined by the eigendecomposition algorithm (used for estimating the universe size and computing the eigenvectors of $\tilde{L}_{\text{nrm}}$) and the linear assignment problem (used for the projection step). The time complexity of the eigendecomposition and optimal linear assignment (e.g., Hungarian algorithm) are respectively $\mathrm{O}(l^{3})$ and $\mathrm{O}(n\,m^{3})$, where $l$ is the number of vertices in the assignment graph, $n$ is the number of views, and $m$ is the size of universe.

In order to improve the speed and scalability of CLEAR, a breadth-first search (BFS), which has the worst computational complexity of $\mathrm{O}(l^{2})$, can be used to first find the connected components of $\tilde{\mathcal{G}}$. The spectrum (i.e., eigenvalues of normalized Laplacian) of $\tilde{\mathcal{G}}$ is then obtained by taking the disjoint union of components’ spectra (similarly eigenvectors are given by zero padding the components’ eigenvectors) [28]. Through this approach, the complexity of the eigendecomposition is reduced to $\mathrm{O}(l_{1}^{3})$, where $l_{1}$ is the number of vertices in the largest connected component of $\tilde{\mathcal{G}}$. In practice, often the association graph consists of many disjoint components (e.g., see examples in Section IX), and the aforementioned procedure considerably improves the runtime and scalability.

The second improvement in speed is achieved by replacing the Hungarian algorithm with the suboptimal sorting strategy. This approach reduces the computational complexity of the projection step to $\mathrm{O}(n\,m^{2}\log(m))$.

VII Applications: Edge-Centric vs. Clique-Centric

In this section, we divide the applications that benefit from solving the multi-view matching problem into two categories, namely edge-centric and clique-centric. It will become clear shortly that making this subtle distinction is crucial for choosing the appropriate evaluation metric for each category.

In edge-centric applications, one ultimately seeks to establish associations between pairs of views (and not all views). In graph terms, this corresponds to seeking individual edges of the association graph (hence the name). For example, using multi-view matching algorithms to associate features between multiple images for estimating relative transformation between the corresponding pair of camera poses [2] belongs to this category. The purpose of using multi-view matching techniques in such applications is to refine the initial noisy associations by incorporating information from multiple views and enforcing cycle consistency. Based on this definition, even a cycle inconsistent set of associations is still a feasible (although erroneous) solution in edge-centric applications. As a result, computing standard metrics such as precision/recall based on individual edges of the association graph is appropriate for evaluating the performance of multi-view association algorithms in such applications.

By contrast, in what we refer to as clique-centric applications, one ultimately seeks to fuse information globally (i.e., across all views) as prescribed by the cliques of the association graph. For example, consider the map fusion problem that arises in single/multi-robot SLAM [4]. After identifying every sighting of each unique landmark in all maps (i.e., encoded in the cliques of a cycle-consistent association graph) via multi-view matching techniques, the corresponding measurements (across all maps) must be fused together in the SLAM back-end to generate the global map. Note that such notion of global fusion is well-defined only if association, as a binary relation on the set of observations, is an equivalence relation.444This mainly refers to transitivity since for all practical purposes in robotics, associations are always reflexive and symmetric. Therefore, unlike edge-centric applications, cycle consistency of associations is a must in clique-centric applications where the observations in each equivalence class are fused together. Cycle-inconsistent solutions must therefore be made cycle consistent before being used in such applications. An implicit and natural way of doing this is via the so-called transitive closure of associations which gives the smallest equivalence relation containing the original associations. In graph terms, this is equivalent to completing each connected component of the association graph into a clique. Thus evaluating such cycle-inconsistent solutions by computing precision/recall based on individual edges can be highly misleading in the case of clique-centric applications. In such cases, precision/recall must be computed after completing the connected components of the association graph (i.e., for the transitive closure).

Note that a single incorrect association only affects local (pairwise) fusions in edge-centric applications, while it may have a catastrophic global impact in clique-centric domains. This is illustrated in Fig. 7 using a simple example. Here the association graph $\mathcal{G}_{1}$ contains only a single incorrect edge drawn in red. Although $\mathcal{G}_{1}$ has a high precision and a high recall for edge-centric applications, it is not cycle consistent and thus does not immediately prescribe a valid solution to clique-centric applications. As mentioned above, for such applications one must first compute the transitive closure of $\mathcal{G}_{1}$. The transitive closure of $\mathcal{G}_{1}$ is given by $\mathcal{G}_{2}$ which performs poorly in terms of precision. Note that each red edge in $\mathcal{G}_{2}$ indicates an incorrect fusion of two observations.

Although CLEAR, by construction, always returns cycle-consistent solutions, as we will see in the following sections several existing algorithms may violate cycle consistency in noisy regimes. It is thus crucial to be aware of the distinction between local (pairwise) and global fusion in order to use the appropriate performance metric in a particular application.

VIII Simulation Results

In this section, we use Monte Carlo analysis with synthetic data to compare CLEAR with several state-of-the-art algorithms across different noise regimes. The aim of these comparisons is to 1) analyze the accuracy of the returned solutions; 2) identify algorithms that violate the cycle consistency or distinctness constraints in high-noise regimes; and 3) evaluate the accuracy of the proposed technique for estimating the universe size.

Algorithms used in our comparisons, which span across three aforementioned domains, include: 1) MatchLift [10] and MatchALS [2] that are based on a convex relaxation; 2) Spectral algorithm [1] extended for partial permutations by Zhou et al. [2], MatchEig [15], and NMFSync [16] that are based on a spectral relaxation; 3) and QuickMatch [3] that is a graph clustering approach.

We consider scenarios with various number of views and observations across different mismatch percentage in the pairwise correspondences. The mismatch in correspondences is introduced by randomly reassigning correct matches to wrong ones according to a uniform distribution. In all comparisons, the universe is set to contain $100$ items, where this value is assumed to be unknown to algorithms and should be estimated. For algorithms that require the knowledge of universe size (all except QuickMatch), the same estimated value obtained for CLEAR from (12) is used.

We report the $\text{F}_{1}$ score, which is commonly used in the literature and is defined as $f\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\frac{2\,p\,r}{p+r}\in[0,1]$, to evaluate the performance of the algorithms. Here, precision $p\in[0,1]$ is defined as the number of correct associations divided by the total number of associations in the output, and recall $r\in[0,1]$ is the number of correct associations in the output divided by the total number of associations in the ground truth. The best performance is achieved when $f=1$ (when $p=q=1$) and the worst when $f=0$ (zero precision and/or zero recall).

In the first comparison, the algorithms are evaluated for different number of views and percentage of mismatch in the input. The observation ratio is fixed at $0.5$; i.e., in each view, $50$ (out of $100$) items of the universe are observed. These items are sampled uniformly at random. For each number of views and mismatch percentage, $10$ Monte Carlo simulations are generated and the average F1 score of the outputs across these simulations is reported in the first row of Fig. 8 (in percentage). In the second comparison (second row in Fig. 8), the number of views in all Monte Carlo simulations is fixed at the value of $n=10$, and results for various observation ratios of universe items and input mismatch percentage are reported. Similar to the first comparison (first row in Fig. 8), each observation ratio indicates the number of items that were observed (i.e., uniformly sampled at random) in a view. For example, observation ratio of $0.2$ indicates that each agent observed $20$ (out of $100$) items of the universe.

Fig. 8 shows that for a fixed observation ratio, as the number of views increases, the F1 score also increases. This indicates that the algorithms are able to leverage the redundancy in observations with the help of the cycle consistency constraint. For the same reason, for a fixed number of views, the F1 score improves as the observation ratio increases.

We also tested the returned solutions for cycle consistency (transitive associations) and distinctness (two observations in a view cannot be associated to each other). The results are displayed using colors in Fig. 8. In particular, here dark green indicates that the (cycle consistent) ground truth was recovered in all Monte Carlo iterations. Light green indicates that the returned solutions satisfied cycle consistency and distinctness, but contained wrong associations in at least one of the simulations. Furthermore, red indicates that, in at least one simulation, the output was not cycle consistent, orange indicates violation of the distinctness constraint, and finally purple indicates violation of both cycle consistency and distinctness constraints.

In addition, Fig. 8 demonstrates that the extended spectral algorithm, MatchEig, and MatchALS may return results that violate the cycle consistency and/or distinctness constraints in moderate to high noise regimes. Recall from Section VII that although a cycle-inconsistent solution may exhibit a high F1 score in terms of individual associations, in clique-centric applications its F1 score can dramatically decrease after completing the connected components of the association graph (i.e., transitive closure). This is demonstrated in Fig. 9 for MatchEIG and MatchALS algorithms (compare Fig. 9 with Fig. 8). For example, the average F1 score of MatchEIG with $10$ views and under $15\%$ mismatch drops from $0.93$ (Fig. 8) to $0.08$ (Fig. 9). As discussed in Section VII, here the F1 score of $0.93$ can be very misleading if the solution obtained by the algorithm is going to be used for fusion in the context of clique-centric applications.

Among the algorithms that do not violate the consistency and distinctness constraints, on average, MatchLift, NMFSync, and CLEAR have the highest F1 scores. The poor performance of QuickMatch is mainly due to the fact that this algorithm was originally designed and tuned for matching image features based on weighted associations, whereas in our setting the associations are binary.555Nonetheless, it is straightforward to generalize CLEAR and other algorithms to the weighted case. In conclusion, synthetic comparisons demonstrate that CLEAR returns cycle consistent solutions with high F1 scores. In the next section, we evaluate the runtime and scalability of the algorithms in real-world examples, where the total number of observations can reach several thousands.

Finally, we compare the estimated size of universe, obtained from (12), with the eigengap method commonly used in the spectral graph clustering literature (see Remark 3). The results are reported in Fig. 10. The number written inside each square is the average of estimated universe sizes (rounded) in the Monte Carlo runs of Fig. 8. The correct universe size is $100$. According to the results depicted in Fig. 10, although both techniques perform equally well under a high signal-to-noise ratio (top two rows in each figure), the proposed approach is more robust to noise and significantly outperforms the standard eigengap heuristic (bottom three rows in each figure).

IX Experimental Results

To further evaluate the accuracy and speed of CLEAR in real-world robotics applications, we consider two scenarios, namely multi-image feature matching and map fusion in landmark-based SLAM. Feature matching datasets have become standard benchmarks for comparing the performance of multi-way data association algorithms. Hence, we report the results on two publicly available standard benchmark datasets, namely Graffiti666 http://www.robots.ox.ac.uk/~vgg/data/data-aff.html and CMU Hotel.777 http://pages.cs.wisc.edu/~pachauri/perm-sync/ The aim of our experimental comparisons is to 1) compare the runtime of algorithms; 2) evaluate the precision/recall for the returned solutions.

IX-A CMU Dataset

The CMU hotel dataset consists of $101$ images. The ground truth provided by this dataset consists of $30$ feature points per image and their correct associations. These feature points are visible across all images, leading to a total of $3030$ features across all images. Due to the large ratio of the number of images ($101$ images) to the number of feature points per image ($30$ features), this dataset represents scenarios where observations have high redundancy. To obtain the input for algorithms, we compute the SIFT descriptor [29] of each feature point using the VLFeat library888http://www.vlfeat.org/ [30]. The standard vl_ubcmatch routine in VLFeat is used to match feature points across image pairs based on the Euclidean distance between their descriptor vectors. By taking this input (as a $3030\times 3030$ aggregate association matrix), each algorithm returns an output which is then compared with the ground truth to evaluate its accuracy. We further record the execution time of each algorithm. All results are based on Matlab implementation of algorithms on a machine with an Intel Core i7-7700K CPU @ 4.20GHz and 16GB RAM.

Fig. 11 shows an example of three images in the CMU hotel sequence, where feature points and their associations across images are shown for the input and the output of four algorithms. Note that the input associations, which are obtained by matching features on image pairs, are cycle inconsistent and contain errors. The output of the algorithms should ideally identify and remove these errors based on the cycle consistency principle.

Fig. 12 reports the precision (i.e., number of correct matches divided by the total number of returned matches) versus the frequency (rate) of the solutions returned by algorithms. The frequency (i.e., the inverse of execution time) indicates the number of times an algorithm can run in one second. Due to the large difference between the runtimes of the algorithms, the frequency axis is scaled logarithmically. The precision of the input is indicated by the orange line on the plot and approximately has the value of $0.92$. Note that this value is calculated based on individual edges and thus is only meaningful for edge-centric applications; see Section VII. Solutions that were not cycle consistent are colored in red. An ideal algorithm should have a high frequency (i.e., small runtime) and a high precision output (i.e., based on individual edges and for edge-centric applications). Among the cycle-consistent algorithms, QuickMatch is the fastest, however, the returned solution does not improve the precision of the input. CLEAR, Spectral, NMFSync, and MatchLift algorithms improve the precision, while CLEAR has a higher frequency: CLEAR is about 3x faster than Spectral, 10x faster than NMFSync, 7500x faster than MatchLift.

The faster runtime of CLEAR is due to 1) the structure of the input association graph, which consists of several disjoint connected components (this graph consists of $81$ connected components, where the largest component has $297$ vertices). This structure is exploited by the proposed eigendecomposition approach, which uses the BFS algorithm to find the spectrum of the graph as the union of its connected components’ spectra. 2) The projection technique, which uses a suboptimal sorting strategy (instead of, e.g., the Hungarian algorithm) to improve the speed while ensuring consistency and distinctness. More specifically, running CLEAR with the Hungarian algorithm results in the same output (i.e., the same value for precision and recall), however, the execution time increases from $0.5$s to $0.7$s.

Fig. 13 reports the precision and recall of returned solutions. An ideal solution simultaneously has high precision and recall. The output of the Spectral algorithm has the highest precision and lowest recall. On the other hand, the output of QuickMatch has the highest recall and lowest precision. In comparison, the output of CLEAR shows a balanced precision versus recall.

We note that the precision and recall of MatchEig after making its solution cycle consistent by completing the association graph’s connected components (Section VII) become $0.67$ and $0.8$, respectively. Similarly, MatchALS’s output after completion takes the precision and recall of $0.73$ and $0.76$, respectively. This sharp drop in precision underlines the importance of taking cycle consistency into account in evaluating multi-view matching algorithms for clique-centric applications.

IX-B Graffiti Dataset

The Graffiti dataset consists of six images, each taken from a different viewpoint of a textured planar wall. Due to the large difference between the viewpoints, this dataset is particularly challenging for feature point detection/matching algorithms (thus, pairwise associations have a lower precision compared to the CMU hotel dataset). The dataset provides ground truth homography transformations between the viewpoints. We use the VLFeat library to extract the SIFT feature points for each image. To obtain the ground truth associations, the provided homography matrices are used to match the extracted features (correct matches must satisfy the planar homography mapping [31, see (5.35)]). To make sure that ground truth associations are error-free, we only take feature points and associations that are cycle consistent across all images and discard the rest. These associations are further visually inspected to ascertain that they do not contain mismatches. The number of feature points retained after this process ranges from $313$ to $657$ per image. The total number of feature points across all images is $3176$. Unlike the CMU hotel dataset, the Graffiti dataset has a small ratio of the number of images to the number of feature points per image. Thus, it represents scenarios where observations have little to no redundancy.

The precision and frequency of algorithms is reported in Fig. 14. Among the cycle-consistent algorithms, QuickMatch is the fastest, however, it does not improve the precision of the input computed based on individual edges and for edge-centric applications (Section VII). CLEAR improves the precision and is considerably faster compared to the other algorithms that improve the input’s precision: about 21x faster than Spectral, 39x faster than NMFSync, 3800x faster than MatchLift.

In the Graffiti dataset, the input association graph consists of $1506$ connected components, where the largest component has $22$ vertices. Running CLEAR with the Hungarian algorithm results in an output with the same value for precision and recall (up to three decimals), however, the execution time of the algorithm increases considerably from $0.92$s to $49.5$s.

The precision and recall of returned solutions are reported in Fig. 15. Among cycle-consistent algorithms, the Spectral algorithm has the highest precision and lowest recall, while QuickMatch has the highest recall and lowest precision. In comparison, CLEAR, MatchLift, and NMFSynch have a balanced precision versus recall. The precision and recall of MatchEig after making its solution cycle consistent (for clique-centric applications) become $0.53$ and $0.69$, respectively. Similarly, MatchALS’s output after completion takes the precision and recall of $0.54$ and $0.69$, respectively. Once again, the difference between these values and those reported in Fig. 15 highlights the importance of taking cycle consistency into account in evaluating multi-view matching algorithms for clique-centric applications.

IX-C Forest Landmark-based SLAM Dataset

Map fusion is an important clique-centric application of the multi-view matching problem in single/multi-robot SLAM [4]. The goal in this problem is to identify unique landmarks across a given set of local maps (created by one or multiple robots) in order to fuse the corresponding measurements in the landmark-based SLAM back-end.999Here, each local map represents a “view” in the multi-view matching problem. In practice, local maps may represent one or multiple frames, and may be built by one or multiple robots. In this section, we report the performance of CLEAR in the context of map fusion based on a SLAM dataset collected in the forest at the NASA Langley Research Center (LaRC) [32]. In this dataset, a single unmanned aerial vehicle equipped with an inertial measurement unit (IMU) and a 2D LIDAR is tasked with autonomously exploring an area under the tree canopy (Fig. 16). The exploration mission lasts $120$ seconds. As the vehicle traverses the forest, it performs LIDAR-inertial odometry by fusing IMU measurements with incremental motion estimates from the iterative closest point algorithm at $40$ Hz. In addition, the vehicle also uses a customized detector to identify trees from the LIDAR scans at a rate of $1$ Hz. The objective is to correctly match and fuse identical tree landmarks detected during the exploration, and subsequently optimize the landmark positions and vehicle trajectory inside a landmark-based SLAM framework.

To obtain the initial pairwise data association, we apply the correspondence graph matching algorithm [33] that associates two sets of landmarks based on their local configurations. Crucially, we note that this process does not use any global pose estimates, and thus is not affected by drift in the LIDAR-inertial odometry. Due to the presence of spurious detections and the lack of informative descriptors (e.g., SIFT), the initial data association matrix (of dimension $1091\times 1091$) contains many mismatches and is not cycle consistent. We thus call CLEAR and other multi-view matching algorithms to achieve cycle consistency. Recall from our discussion in Section VII that map fusion is inherently a clique-centric application. Therefore, we make any inconsistent data associations cycle consistent by completing the connected components in the association graph (Section VII). In addition, we also introduce a baseline algorithm that directly completes the connected components in the input associations.

Since ground truth data association is not available, we adopt the following alternative performance metrics. A pair of associated trees is classified as either a definite negative or a potential positive, based on whether their distance as estimated by the LIDAR-inertial odometry is higher than a threshold of $2$ m. We note that these definitions are precise assuming that the threshold value of $2$ m accounts for the drift in the LIDAR-inertial odometry.101010Since the vehicle is flying at a low speed ($2$ m/s) for a relative short amount of time $120$ s, we expect the estimation drift at any time is reasonably bounded. Since the number of definite negatives (denoted by DN) is an underestimate of the true number of mismatches, and the number of potential positives (denoted by PP) is an overestimate of the true number of correct matches, we can further calculate an upper bound on the true precision as follows,

[TABLE]

We note that for landmark-based SLAM, the number of definite negatives (DN) is particularly important, since it is well known that any false data association could inflict catastrophic impact on the final solution. Therefore, an ideal data association should contain no definite negatives, or equivalently achieve a value of $100\%$ for $\bar{\text{P}}$ (upper bound on precision).

Fig. 17 visualizes the data associations returned by each algorithm in the world frame. All definite negatives are highlighted in red. The solutions of MatchALS and MatchEIG are not cycle consistent initially, and are made cycle consistent by completing the connected components. The solution of the spectral algorithm is omitted, as it contains significantly more mismatches due to its sensitivity in estimating the universe size. Table II shows the complete set of quantitative results and Fig. 18 shows the precision and frequency of each evaluated algorithm.

Due to the existence of mismatches in the input associations, the baseline algorithm which directly completes the connected components yields more than $25000$ definite negatives; see Fig. 17(a). In contrast, most other algorithms are able to significantly reduce the number of definite negatives. Among these algorithms, CLEAR and MatchLift nearly eliminate all definite negatives; see Table II. However, MatchLift requires $124.7$ s to converge while CLEAR only takes $0.084$ s. The superior speed of CLEAR thus makes the algorithm favorable for real-time SLAM applications. On the other hand, we note that the output of CLEAR still contains a few definite negatives; see Fig. 17(g). This is undesirable for landmark-based SLAM, as any incorrect fusion of landmarks could inflict catastrophic impact on the final SLAM solution. In practice, these mismatches can be filtered out by removing clusters of small size from the returned solution. For example, Fig. 17(h) shows the resulting association after removing clusters of size smaller than four from the output of CLEAR. After this post-processing step, the final association is accurate and can be used by any SLAM back-end to solve for the vehicle trajectory and landmark positions.

Fig. 19 demonstrates the results of landmark-based SLAM using the data association returned by each algorithm. Prior to optimization, we apply the same post-processing procedure described earlier, by removing clusters of size smaller than four from each data association. Subsequently, we initialize a single tree for each remaining cluster in the fused map. All tree positions and vehicle poses are then jointly optimized using g2o [34]. We note that Fig. 19 mainly provides a qualitative comparison of the trajectory estimates. Intuitively, we expect that SLAM trajectories that are discontinuous are likely to be wrong due to false data associations. These include the trajectories optimized with the baseline, NMFSync, MatchALS, MatchEIG, and QuickMatch. While CLEAR and MatchLift produce similar results, CLEAR is more than $1000$ times faster as indicated by the results in Table II.

Finally, we note that some quantitative results presented in this section are different from those obtained in earlier comparisons based on synthetic data. For example, we observed that algorithms such as NMFSync perform better in simulation. A major cause of this discrepancy is the noise model. In our synthetic data, the input is solely corrupted by mismatches that reassign correct matches to wrong ones. In the forest dataset, however, the input is corrupted by both mismatches and a significant number of missing correct associations, thus further reducing the signal-to-noise ratio.

X Conclusion

Data association across multiple views is a fundamental problem in robotic applications. Traditionally, this problem is decomposed into a sequence of pairwise subproblems. Multi-view matching algorithms can leverage observation redundancy to improve the accuracy of pairwise associations. However, the use of these algorithms in robotic applications is often prohibited by their high computational complexity, as well as critical issues such as cycle inconsistency and high number of mismatches which may have catastrophic consequences.

To address these critical challenges, we presented CLEAR, an algorithm that leverages the natural graphical representation of the multi-view association problem. CLEAR uses a spectral graph clustering technique, which is uniquely tailored to solve this problem in a computationally efficient manner. Empirical results based on extensive synthetic and experimental evaluations demonstrated that CLEAR outperforms the state-of-the-art algorithms in terms of both accuracy and speed. This general framework can provide significant improvements in the accuracy and efficiency of data association in many applications such as metric/semantic SLAM, multi-object tracking, and multi-view point cloud registration that traditionally rely on pairwise matchings.

Acknowledgments

This work was supported in part by NASA Convergent Aeronautics Solutions project Design Environment for Novel Vertical Lift Vehicles (DELIVER), by ONR under BRC award N000141712072, and by ARL DCIST under Cooperative Agreement Number W911NF-17-2-0181. We would also like to thank Florian Bernard, Kostas Daniilidis, Spyros Leonardos, Stephen Phillips, Roberto Tron, and Xiaowei Zhou for their valuable insights that considerably improved this work.