Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

Bandeira, Afonso S.; Boumal, Nicolas; Singer, Amit

doi:10.1007/s10107-016-1059-6

Cited by 117 publications

(249 citation statements)

References 74 publications

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“…Similar to our case, the tightness of the relaxation, i.e. obtaining rank-1 solutions from semidefinite relaxations, is also observed in several different SDR formulations (see, e.g., [15,5] and the survey [71]). …”

Section: Camera Location Estimation In Sfmsupporting

confidence: 80%

Stable Camera Motion Estimation Using Convex Programming

Özyeşil¹,

Singer²,

Basri³

2015

SIAM J. Imaging Sci.

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113

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Abstract. We study the inverse problem of estimating n locations t 1 , t 2 , . . . , tn (up to global scale, translation and negation) in R d from noisy measurements of a subset of the (unsigned) pairwise lines that connect them, that is, from noisy measurements of ± t i −t j t i −t j 2 for some pairs (i, j) (where the signs are unknown). This problem is at the core of the structure from motion (SfM) problem in computer vision, where the t i 's represent camera locations in R 3 . The noiseless version of the problem, with exact line measurements, has been considered previously under the general title of parallel rigidity theory, mainly in order to characterize the conditions for unique realization of locations. For noisy pairwise line measurements, current methods tend to produce spurious solutions that are clustered around a few locations. This sensitivity of the location estimates is a well-known problem in SfM, especially for large, irregular collections of images.In this paper we introduce a semidefinite programming (SDP) formulation, specially tailored to overcome the clustering phenomenon. We further identify the implications of parallel rigidity theory for the location estimation problem to be well-posed, and prove exact (in the noiseless case) and stable location recovery results. We also formulate an alternating direction method to solve the resulting semidefinite program, and provide a distributed version of our formulation for large numbers of locations. Specifically for the camera location estimation problem, we formulate a pairwise line estimation method based on robust camera orientation and subspace estimation. Lastly, we demonstrate the utility of our algorithm through experiments on real images.Key words. Structure from motion, parallel rigidity, semidefinite programming, convex relaxation, alternating direction method of multipliers AMS subject classifications. 68T45, 52C25, 90C22, 90C251. Introduction. Global positioning of n objects from partial information about their relative locations is prevalent in many applications spanning fields such as sensor network localization [8,58,16,18], structural biology [31], and computer vision [27,9]. A well-known instance that attracted much attention from both the theoretical and algorithmic perspectives is that of estimating the locations t 1 , t 2 , . . . , t n ∈ R d from their pairwise Euclidean distances t i − t j 2 . In this case, the large body of literature in rigidity theory (cf. [4, 54]) provides conditions under which the localization is unique given a set of noiseless distance measurements. Also, much progress has been made with algorithms that estimate positions from noisy distances, starting with classical multidimensional scaling [48] to the more recent semidefinite programming (SDP) approaches (see, e.g., [8,7]).Here we consider a different global positioning problem, in which the locations t 1 , . . . , t n need to be estimated from a subset of (potentially noisy) measurements of the pairwise lines that connect them (see Figure 1.1 for a ...

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Section: Camera Location Estimation In Sfmsupporting

confidence: 80%

Stable Camera Motion Estimation Using Convex Programming

Özyeşil¹,

Singer²,

Basri³

2015

SIAM J. Imaging Sci.

Self Cite

113

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“…Uð1Þ synchronization has been used as a model for clock synchronization over networks (18,19). It is also closely related to the phase-retrieval problem in signal processing (20)(21)(22).…”

Section: Significancementioning

confidence: 99%

Phase transitions in semidefinite relaxations

Javanmard

Montanari

Ricci-Tersenghi

2016

Proc. Natl. Acad. Sci. U.S.A.

108

126

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Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large-scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that when the statistical noise is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several detection thresholds, as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems. Modern datasets pose new challenges to this centuries-old framework. On one hand, high-dimensional applications require the simultaneous estimation of millions of parameters. Examples span genomics (2), imaging (3), web services (4), and so on. On the other hand, the unknown object to be estimated has often a combinatorial structure: In clustering we aim at estimating a partition of the data points (5). Network analysis tasks usually require identification of a discrete subset of nodes in a graph (6, 7). Parsimonious data explanations are sought by imposing combinatorial sparsity constraints (8).There is an obvious tension between the above requirements. Although efficient algorithms are needed to estimate a large number of parameters, the maximum likelihood (ML) method often requires the solution of NP-hard (nondeterministic polynomial-time hard) combinatorial problems. A flourishing line of work addresses this conundrum by designing effective convex relaxations of these combinatorial problems (9-11).Unfortunately, the statistical properties of such convex relaxations are well understood only in a few cases [compressed sensing being the most important success story (12-14)]. In this paper we use tools from statistical mechanics to develop a precise picture of the behavior of a class of semidefinite programming relaxations. Relaxations of this type appear to be surprisingly effective in a variety of problems ranging from clustering to graph synchronization. For the sake of concreteness we will focus on three specific problems. Z 2 SynchronizationIn the general synchronization problem, we aim at estimating x 0,1 , x 0,2 , . . . , x 0,n , which are unknown elements of ...

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“…6 To be concrete, in the real case the entries are N .0; 1/, and in the complex case the real and imaginary parts of each entry are N .0; 1=2/. The noise W is a Gaussian random matrix drawn from the GOE, GUE, or GSE, depending on whether is of real type, complex type, or quaternionic type, respectively.…”

Section: Gaussian Observation Modelmentioning

confidence: 99%

“…Results are known for slower convex programs [6]. Boumal's algorithm now iterates v f .Y v/, where f divides each entry by its norm, thus projecting to the unit circle.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Message‐Passing Algorithms for Synchronization Problems over Compact Groups

Perry

Wein

Bandeira

et al. 2018

Comm Pure Appl Math

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Various alignment problems arising in cryo-electron microscopy, community detection, time synchronization, computer vision, and other fields fall into a common framework of synchronization problems over compact groups such as Z=L, U.1/, or SO.3/. The goal in such problems is to estimate an unknown vector of group elements given noisy relative observations. We present an efficient iterative algorithm to solve a large class of these problems, allowing for any compact group, with measurements on multiple "frequency channels" (Fourier modes, or more generally, irreducible representations of the group). Our algorithm is a highly efficient iterative method following the blueprint of approximate message passing (AMP), which has recently arisen as a central technique for inference problems such as structured low-rank estimation and compressed sensing. We augment the standard ideas of AMP with ideas from representation theory so that the algorithm can work with distributions over general compact groups. Using standard but nonrigorous methods from statistical physics, we analyze the behavior of our algorithm on a Gaussian noise model, identifying phases where we believe the problem is easy, (computationally) hard, and (statistically) impossible. In particular, such evidence predicts that our algorithm is information-theoretically optimal in many cases, and that the remaining cases exhibit statistical-to-computational gaps.

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Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

Cited by 117 publications

References 74 publications

Stable Camera Motion Estimation Using Convex Programming

Stable Camera Motion Estimation Using Convex Programming

Phase transitions in semidefinite relaxations

Message‐Passing Algorithms for Synchronization Problems over Compact Groups

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