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Vector diffusion maps and the connection laplacian
 CComm. Pure Appl. Math
"... Abstract. We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as L ..."
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Cited by 48 (13 self)
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Abstract. We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on (or near) a low dimensional manifold M d embedded in R p, we prove the relation between VDM and the connectionLaplacian operator for vector fields over the manifold. Key words. Dimensionality reduction, vector fields, heat kernel, parallel transport, local principal component analysis, alignment. 1. Introduction. Apopularwaytodescribethe
ThreeDimensional Structure Determination from Common Lines in CryoEM by Eigenvectors and Semidefinite Programming
, 2011
"... The cryoelectron microscopy reconstruction problem is to find the threedimensional (3D) structure of a macromolecule given noisy samples of its twodimensional projection images at unknown random directions. Present algorithms for finding an initial 3D structure model are based on the “angular r ..."
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Cited by 33 (17 self)
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The cryoelectron microscopy reconstruction problem is to find the threedimensional (3D) structure of a macromolecule given noisy samples of its twodimensional projection images at unknown random directions. Present algorithms for finding an initial 3D structure model are based on the “angular reconstitution ” method in which a coordinate system is established from three projections, and the orientation of the particle giving rise to each image is deduced from common lines among the images. However, a reliable detection of common lines is difficult due to the low signaltonoise ratio of the images. In this paper we describe two algorithms for finding the unknown imaging directions of all projections by minimizing global selfconsistency errors. In the first algorithm, the minimizer is obtained by computing the three largest eigenvectors of a specially designed symmetric matrix derived from the common lines, while the second algorithm is based on semidefinite programming (SDP). Compared with existing algorithms, the advantages of our algorithms are fivefold: first, they accurately estimate all orientations at very low commonline detection rates; second, they are extremely fast, as they involve only the computation of a few top eigenvectors or a sparse SDP; third, they are nonsequential and use the information in all common lines at once; fourth, they are amenable to a rigorous mathematical analysis using spectral analysis and random matrix theory; and finally, the algorithms are optimal in the sense that they reach the information theoretic Shannon bound up to a constant for an idealized probabilistic model.
Viewing Angle Classification of CryoElectron Microscopy Images Using Eigenvectors
, 2011
"... The cryoelectron microscopy (cryoEM) reconstruction problem is to find the threedimensional structure of a macromolecule given noisy versions of its twodimensional projection images at unknown random directions. We introduce a new algorithm for identifying noisy cryoEM images of nearby viewing ..."
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Cited by 28 (16 self)
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The cryoelectron microscopy (cryoEM) reconstruction problem is to find the threedimensional structure of a macromolecule given noisy versions of its twodimensional projection images at unknown random directions. We introduce a new algorithm for identifying noisy cryoEM images of nearby viewing angles. This identification is an important first step in threedimensional structure determination of macromolecules from cryoEM, because once identified, these images can be rotationally aligned and averaged to produce “class averages” of better quality. The main advantage of our algorithm is its extreme robustness to noise. The algorithm is also very efficient in terms of running time and memory requirements, because it is based on the computation of the top few eigenvectors of a specially designed sparse Hermitian matrix. These advantages are demonstrated in numerous numerical experiments.
A Cheeger inequality for the graph connection Laplacian
, 2012
"... 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) sy ..."
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Cited by 24 (14 self)
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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.
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 23 (14 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.
Phase retrieval with polarization
 SIAM J. ON IMAGING SCI
, 2013
"... In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and ..."
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Cited by 22 (5 self)
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In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [14]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternatingprojectionstype procedure.
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 22 (9 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
Global Motion Estimation from Point Matches
"... Abstract—Multiview structure recovery from a collection of images requires the recovery of the positions and orientations of the cameras relative to a global coordinate system. Our approach recovers camera motion as a sequence of two global optimizations. First, pairwise Essential Matrices are used ..."
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Cited by 15 (4 self)
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Abstract—Multiview structure recovery from a collection of images requires the recovery of the positions and orientations of the cameras relative to a global coordinate system. Our approach recovers camera motion as a sequence of two global optimizations. First, pairwise Essential Matrices are used to recover the global rotations by applying robust optimization using either spectral or semidefinite programming relaxations. Then, we directly employ feature correspondences across images to recover the global translation vectors using a linear algorithm based on a novel decomposition of the Essential Matrix. Our method is efficient and, as demonstrated in our experiments, achieves highly accurate results on collections of real images for which ground truth measurements are available. Keywordsstructure from motion; 3D reconstruction; camera motion estimation; convex relaxation; linear estimation I.
Decoding binary node labels from censored edge measurements: Phase transition and efficient recovery
, 2014
"... We consider the problem of clustering a graphG into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y = BGx⊕Z, where BG is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a ..."
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Cited by 12 (6 self)
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We consider the problem of clustering a graphG into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y = BGx⊕Z, where BG is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a uniform prior) and Z is a noise vector with Bernoulli(ε) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of x is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n)/n for ErdősRényi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by α = np / log(n), it is shown that exact recovery is possible if and only if α> 2/(1 − 2ε)2 + o(1/(1 − 2ε)2). In other words, 2/(1 − 2ε)2 is the information theoretic threshold for exact recovery at lowSNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph G, defining the degree rate as α = d / log(n), where d is the minimum degree of the graph, it is shown that the proposed method achieves the rate α> 4((1 + λ)/(1 − λ)2)/(1 − 2ε)2 + o(1/(1 − 2ε)2), where 1 − λ is the spectral gap of the graph G. A preliminary version of this paper appeared in ISIT 2014 [ABBS14].