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21
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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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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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
Decoding binary node labels from censored edge measurements: Phase transition and efficient recovery. available at arXiv:1404.4749 [cs.IT
, 2014
"... Abstract. 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 ..."
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Cited by 9 (4 self)
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Abstract. 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]. 1.
Convex recovery from interferometric measurements. arXiv preprint arXiv:1307.6864
, 2013
"... This note formulates a deterministic recovery result for vectors x from quadratic measurements of the form (Ax)i(Ax)j for some leftinvertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a wellconnected graph. One possible way of obtaining the ..."
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This note formulates a deterministic recovery result for vectors x from quadratic measurements of the form (Ax)i(Ax)j for some leftinvertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a wellconnected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a dataweighted graph Laplacian. Such quadratic measurements have found applications in phase retrieval, angular synchronization, and more recently interferometric waveform inversion. Acknowledgments. The authors would like to thank Amit Singer for interesting discussions. 1
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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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.
Global registration of multiple point clouds using semidefinite programming. arXiv:1306.5226 [cs.CV
, 2013
"... ABSTRACT. Consider N points in R d and M local coordinate systems that are related through unknown rigid transforms. For each point we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordin ..."
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Cited by 8 (4 self)
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ABSTRACT. Consider N points in R d and M local coordinate systems that are related through unknown rigid transforms. For each point we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The leastsquares formulation, though nonconvex, has a well known closedform solution for the case M = 2 (based on the singular value decomposition). However, no closed form solution is known for M ≥ 3. In this paper, we propose a semidefinite relaxation of the leastsquares formulation, and prove conditions for exact and stable recovery for both this relaxation and for a previously proposed spectral relaxation. In particular, using results from rigidity theory and the theory of semidefinite programming, we prove that the semidefinite relaxation can guarantee recovery under more adversarial measurements compared to the spectral counterpart. We perform numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the semidefinite program (i.e., we are able to solve the original nonconvex problem) up to a certain noise threshold, and (b) the semidefinite program performs significantly better than spectral and manifoldoptimization methods, particularly at large noise levels.
Linear inverse problems on ErdősRényi graphs: Informationtheoretic limits and efficient recovery
"... Abstract—This paper considers 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. T ..."
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Cited by 6 (4 self)
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Abstract—This paper considers 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 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. Full version available in [1]. I.
Alternating Projection, Ptychographic Imaging and Phase Synchronization. ArXiv eprints
, 2014
"... Abstract. We demonstrate necessary and sufficient conditions of the global convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychographic imaging problem, we discuss phase synchronization and connection graph Laplacian, and ..."
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Abstract. We demonstrate necessary and sufficient conditions of the global convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychographic imaging problem, we discuss phase synchronization and connection graph Laplacian, and show how to construct an accurate initial guess to accelerate convergence speed to handle the big imaging data in the coming new light source era. 1.
Cramérrao bounds for synchronization of rotations
 CoRR
"... Synchronization of rotations is the problem of estimating a set of rotations Ri ∈ SO(n), i = 1... N based on noisy measurements of relative rotations RiR ⊤ j. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchron ..."
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Synchronization of rotations is the problem of estimating a set of rotations Ri ∈ SO(n), i = 1... N based on noisy measurements of relative rotations RiR ⊤ j. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zeromean isotropic noise, and we develop tools for Gaussianlike as well as heavytail types of noise in particular. As a main contribution, we derive the CramérRao bounds of synchronization, that is, lowerbounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchorfree scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussianlike noise. Synchronization of rotations, estimation on manifolds, estimation on graphs, graph Laplacian, Fisher information, CramérRao bounds, distributions on the rotation group, Langevin. 2000 Math Subject Classification: 62F99, 94C15, 22C05, 05C12, 1