Angular synchronization by eigenvectors and semidefinite programming

Singer, Amit

doi:10.1016/j.acha.2010.02.001

Cited by 351 publications

(500 citation statements)

References 47 publications

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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.

111

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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Section: Significancementioning

confidence: 99%

Phase transitions in semidefinite relaxations

Javanmard

Montanari

Ricci-Tersenghi

2016

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

111

126

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“…A theoretical study of these relaxations is an interesting topic beyond the scope of this manuscript. Finally, we remark that similar relaxations appear in a different problem of angular synchronization, as studied by Singer [42].…”

Section: A Finding the Minimizer Ofmentioning

confidence: 99%

Vectorial Phase Retrieval of 1-D Signals

Raz

Dudovich

Nadler

2013

IEEE Trans. Signal Process.

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Abstract-Reconstruction of signals from measurements of their spectral intensities, also known as the phase retrieval problem, is of fundamental importance in many scientific fields. In this paper we present a novel framework, denoted as vectorial phase retrieval, for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference. We show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal. First, we prove that for compactly supported signals, in the absence of measurement noise, this new setup admits a unique solution. Next, we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it. Finally, we illustrate via simulations, that our algorithm can accurately reconstruct signals even at considerable noise levels.Index Terms-Convex relaxation, 1-D phase retrieval, signal recovery from modulus Fourier measurements, statistical model selection.

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“…In this schematic, registration via angular synchronization is trivial, as the pairwise measurements contain no noise. However, the algorithm can register data sets even when many of the pairwise measurements are inaccurate (Singer, 2011).…”

Section: Vector Diffusion Maps For Registration and Temporal Orderingmentioning

confidence: 99%

“…Vector diffusion maps combine two algorithms -angular synchronization (Singer, 2011) for image registration, and diffusion maps (Coifman et al, 2005) for extracting intrinsic low-dimensional structure in data -into a single computation. We will use the algorithm to register images of developing tissues with respect to planar rotations, as well as uncover the main direction of variability after removing rotational symmetries.…”

Section: Vector Diffusion Maps For Registration and Temporal Orderingmentioning

confidence: 99%

Temporal ordering and registration of images in studies of developmental dynamics

Dsilva

Lim

et al. 2015

Development

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Progress of development is commonly reconstructed from imaging snapshots of chemical or mechanical processes in fixed tissues. As a first step in these reconstructions, snapshots must be spatially registered and ordered in time. Currently, image registration and ordering are often done manually, requiring a significant amount of expertise with a specific system. However, as the sizes of imaging data sets grow, these tasks become increasingly difficult, especially when the images are noisy and the developmental changes being examined are subtle. To address these challenges, we present an automated approach to simultaneously register and temporally order imaging data sets. The approach is based on vector diffusion maps, a manifold learning technique that does not require a priori knowledge of image features or a parametric model of the developmental dynamics. We illustrate this approach by registering and ordering data from imaging studies of pattern formation and morphogenesis in three model systems. We also provide software to aid in the application of our methodology to other experimental data sets.

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Angular synchronization by eigenvectors and semidefinite programming

Cited by 351 publications

References 47 publications

Phase transitions in semidefinite relaxations

Phase transitions in semidefinite relaxations

Vectorial Phase Retrieval of 1-D Signals

Temporal ordering and registration of images in studies of developmental dynamics

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