The Geometry of Synchronization Problems and Learning Group Actions

Gao, Tingran; Brodzki, Jacek; Mukherjee, Sayan

doi:10.1007/s00454-019-00100-2

Cited by 16 publications

(16 citation statements)

References 152 publications

(231 reference statements)

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“…Among the most prominent methods to date are spectral methods [Sin11,BSS11], semidefinite relaxations [BCSZ14, BCS15, ABBS14, BBS16, JMRT16, BBV16], methods based on Approximate Message Passing [PWBM16] and other modified power methods [Bou16,CC16]. Synchronization also enjoys many interesting connections with geometry (see, e.g., [GBM16]).…”

Section: Prior Work: the Synchronization Approachmentioning

confidence: 99%

Optimal rates of estimation for multi-reference alignment

2020

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In this paper, we establish optimal rates of adaptive estimation of a vector in the multi-reference alignment model, a problem with important applications in fields such as signal processing, image processing, and computer vision, among others. We describe how this model can be viewed as a multivariate Gaussian mixture model under the constraint that the centers belong to the orbit of a group. This enables us to derive matching upper and lower bounds that feature an interesting dependence on the signal-to-noise ratio of the model. Both upper and lower bounds are articulated around a tight local control of Kullback-Leibler divergences that showcases the central role of moment tensors in this problem.

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Section: Prior Work: the Synchronization Approachmentioning

confidence: 99%

Optimal rates of estimation for multi-reference alignment

2020

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“…The PPM is applicable to other combinatorial problems beyond joint alignment. We present here an example called joint graph matching [18,33,43,52,63,69]. Consider a collection of n images, each containing m feature points, and suppose that there exists a one-to-one correspondence between the feature points in any pair of images.…”

Section: Joint Graph Matchingmentioning

confidence: 99%

The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differences

Chen

Candès

2018

Comm Pure Appl Math

115

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Various applications involve assigning discrete label values to a collection of objects based on some pairwise noisy data. Due to the discrete—and hence nonconvex—structure of the problem, computing the optimal assignment (e.g., maximum‐likelihood assignment) becomes intractable at first sight. This paper makes progress towards efficient computation by focusing on a concrete joint alignment problem; that is, the problem of recovering n discrete variables xi ∊ {1, …, m}, 1 ≤ i ≤ n, given noisy observations of their modulo differences {xi — xj mod m}. We propose a low‐complexity and model‐free nonconvex procedure, which operates in a lifted space by representing distinct label values in orthogonal directions and attempts to optimize quadratic functions over hypercubes. Starting with a first guess computed via a spectral method, the algorithm successively refines the iterates via projected power iterations. We prove that for a broad class of statistical models, the proposed projected power method makes no error—and hence converges to the maximum‐likelihood estimate—in a suitable regime. Numerical experiments have been carried out on both synthetic and real data to demonstrate the practicality of our algorithm. We expect this algorithmic framework to be effective for a broad range of discrete assignment problems.© 2018 Wiley Periodicals, Inc.

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“…There are more sophisticated variants of these ideas, with applications passing beyond finding consistent rankings or orderings. Recent work of Gao et al [49] gives a cohomological and Hodge-theoretic approach to synchronization problems over networks based on pairwise nodal data in the presence of noise. Singer and collaborators [85,86] have published several works on cryo electron miscroscopy that is, in essence, a cohomological approach to finding consistent solutions to pairwise-compared data over a network.…”

Section: Cohomology and Rankingmentioning

confidence: 99%

Homological algebra and data

Ghrist

2018

IAS/Park City Mathematics Series

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These lectures are a quick primer on the basics of applied algebraic topology with emphasis on applications to data. In particular, the perspectives of (elementary) homological algebra, in the form of complexes and co/homological invariants are sketched. Beginning with simplicial and cell complexes as a means of enriching graphs to higher-order structures, we define simple algebraic topological invariants, such as Euler characteristic. By lifting from complexes of simplices to algebraic complexes of vector spaces, we pass to homology as a topological compression scheme. Iterating this process of expanding to sequences and compressing via homological algebra, we define persistent homology and related theories, ending with a simple approach to cellular sheaves and their cohomology. Throughout, an emphasis is placed on expressing homological-algebraic tools as the natural evolution of linear algebra. Category-theoretic language (though more natural and expressive) is deemphasized, for the sake of access. Along the way, sample applications of these techniques are sketched, in domains ranging from neuroscience to sensing, image analysis, robotics, and computation.

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The Geometry of Synchronization Problems and Learning Group Actions

Cited by 16 publications

References 152 publications

Optimal rates of estimation for multi-reference alignment

Optimal rates of estimation for multi-reference alignment

The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differences

Homological algebra and data

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