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ANGULAR SYNCHRONIZATION BY EIGENVECTORS AND SEMIDEFINITE PROGRAMMING: ANALYSIS AND APPLICATION TO CLASS AVERAGING IN CRYOELECTRON MICROSCOPY
, 905
"... Abstract. The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1,..., θn from m noisy measurements of their offsets θi − θj mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements ..."
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Cited by 43 (17 self)
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Abstract. The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1,..., θn from m noisy measurements of their offsets θi − θj mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in [0,2π) and carry no information on the true offsets. We introduce an efficient recovery algorithm for the unknown angles from the top eigenvector of a specially designed Hermitian matrix. The eigenvector method is extremely stable and succeeds even when the number of outliers is exceedingly large. For example, we successfully estimate n = 400 angles from a full set of m = `400 ´ offset measurements of which 90 % are outliers in less than a second 2 on a commercial laptop. We use random matrix theory to prove that the eigenvector method q gives
Sensor network localization by eigenvector synchronization over the Euclidean group
 In press
"... We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding and aligning uniquely realizable subsets of neighboring sensors called patches. In the noisefree case, each patch agrees with its global ..."
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Cited by 25 (15 self)
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We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding and aligning uniquely realizable subsets of neighboring sensors called patches. In the noisefree case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. The algorithm is scalable as the number of nodes increases, and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity and running time. While our approach is applicable to higher dimensions, in the current paper we focus on the two dimensional case.
A Cheeger inequality for the graph connection laplacian. available online
, 2012
"... Abstract. The O(d) Synchronization problem consists of estimating a set of n unknown orthogonal d × d matrices O1,..., On from noisy measurements of a subset of the pairwise ratios OiO −1 j. We formulate and prove a Cheegertype inequality that relates a measure of how well it is possible to solve t ..."
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Cited by 23 (13 self)
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Abstract. The O(d) Synchronization problem consists of estimating a set of n unknown orthogonal d × d matrices O1,..., On from noisy measurements of a subset of the pairwise ratios OiO −1 j. We formulate and prove a Cheegertype inequality that relates a measure of how well it is possible to solve the O(d) synchronization problem with the spectra of an operator, the graph Connection Laplacian. We also show how this inequality provides a worst case performance guarantee for a spectral method to solve this problem.
Representation theoretic patterns in three dimensional cryoelectron macroscopy I  The Intrinsic reconstitution algorithm
"... Abstract. In this paper, we continue to develop the representation theoretic setup for 3D cryoelectron microscopy (cryoEM) that was initiated in [7]. In particular, we provide a complete spectral analysis of the local parallel transport operator on the two dimensional sphere. This is then used to ..."
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Cited by 20 (8 self)
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Abstract. In this paper, we continue to develop the representation theoretic setup for 3D cryoelectron microscopy (cryoEM) that was initiated in [7]. In particular, we provide a complete spectral analysis of the local parallel transport operator on the two dimensional sphere. This is then used to prove the admissibility (correctness) and the numerical stability of the intrinsic classification algorithm for identifying raw projection images of similar viewing directions in cryoEM, that was recently introduced in [9]. This preliminary classification is of fundamental importance in determining the three dimensional structure of macromolecules from cryoEM images. The goal in cryoEM is to determine the 3D structure of a molecule from noisy projection images taken at unknown random orientations by an electron microscope, i.e., a random Computational Tomography (CT). Determining 3D structures of
EXACT AND STABLE RECOVERY OF ROTATIONS FOR ROBUST SYNCHRONIZATION
, 1211
"... Abstract. The synchronization problem over the special orthogonal group SO(d) consists of estimating a set of unknown rotations R1, R2,..., Rn from noisy measurements of a subset of their pairwise ratios R −1 i Rj. The problem has found applications in computer vision, computer graphics, and sensor ..."
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Cited by 20 (7 self)
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Abstract. The synchronization problem over the special orthogonal group SO(d) consists of estimating a set of unknown rotations R1, R2,..., Rn from noisy measurements of a subset of their pairwise ratios R −1 i Rj. The problem has found applications in computer vision, computer graphics, and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of MaxCut. The contribution of this paper is threefold: First, we introduce a robust penalty function involving the sum of unsquared deviations and derive a relaxation that leads to a convex optimization problem; Second, we apply the alternating direction method to minimize the penalty function; Finally, under a specific model of the measurement noise and the measurement graph, we prove that the rotations are exactly and stably recovered, exhibiting a phase transition behavior in terms of the proportion of noisy measurements. Numerical simulations confirm the phase transition behavior for our method as well as its improved accuracy compared to existing methods. Key words. Synchronization of rotations; least unsquared deviation; semidefinite relaxation; and alternating direction method 1. Introduction. The
SPECTRAL CONVERGENCE OF THE CONNECTION LAPLACIAN FROM RANDOM SAMPLES
, 1306
"... ABSTRACT. Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are extremely useful for manifold learning. It was previously shown by Belkin and Niyogi [4] that the eigenvectors and eigenvalues of the graph Lapla ..."
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Cited by 9 (4 self)
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ABSTRACT. Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are extremely useful for manifold learning. It was previously shown by Belkin and Niyogi [4] that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the LaplaceBeltrami operator of the manifold in the limit of infinitely many uniformly sampled data points. Recently, we introduced Vector Diffusion Maps and showed that the Connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other Connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many random samples. Our results for spectral convergence also hold in the case where the data points are sampled from a nonuniform distribution, and for manifolds with and without boundary. 1.
Computing steerable principal components of a large set of images and their rotations
 IEEE Trans. Image Process
"... We present here an efficient algorithm to compute the principal component analysis (PCA) of a large image set consisting of images and, for each image, the set of its uniform rotations in the plane. We do this by pointing out the block circulant structure of the covariance matrix and utilizing that ..."
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Cited by 8 (2 self)
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We present here an efficient algorithm to compute the principal component analysis (PCA) of a large image set consisting of images and, for each image, the set of its uniform rotations in the plane. We do this by pointing out the block circulant structure of the covariance matrix and utilizing that structure to compute its eigenvectors. We also demonstrate the advantages of this algorithm over similar ones with numerical experiments. Although it is useful in many settings, we illustrate the specific application of the algorithm to the problem of cryoelectron microscopy.
The Spectrum of Random Innerproduct Kernel Matrices
, 1202
"... Abstract: We consider nbyn matrices whose (i,j)th entry is f(X T i Xj), where X1,...,Xn are i.i.d. standard Gaussian random vectors in R p, and f is a realvalued function. The eigenvalue distribution of these random kernel matrices is studied at the “large p, large n ” regime. It is shown that, ..."
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Cited by 6 (0 self)
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Abstract: We consider nbyn matrices whose (i,j)th entry is f(X T i Xj), where X1,...,Xn are i.i.d. standard Gaussian random vectors in R p, and f is a realvalued function. The eigenvalue distribution of these random kernel matrices is studied at the “large p, large n ” regime. It is shown that, when p,n → ∞ and p/n = γ which is a constant, and f is properly scaled so that Var(f(X T i Xj)) is O(p −1), the spectral density converges weakly to a limiting density on R. The limiting density is dictated by a cubic equation involving its Stieltjes transform. While for smooth kernel functions the limiting spectral density has been previously shown to be the MarcenkoPastur distribution, our analysis is applicable to nonsmooth kernel functions, resulting in a new family of limiting densities.
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Cited by 6 (1 self)
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.