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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 Cheeger-type 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 Cheeger-type 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.
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 Max-Cut. The contribution of this paper is three-fold: 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 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 least-squares formulation, though non-convex, has a well known closed-form 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 least-squares 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 non-convex problem) up to a certain noise threshold, and (b) the semidefinite program performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.
Cramér-rao 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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Cited by 4 (4 self)
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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 zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cramér-Rao bounds of synchronization, that is, lower-bounds 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 anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise. Synchronization of rotations, estimation on manifolds, estimation on graphs, graph Laplacian, Fisher information, Cramér-Rao bounds, distributions on the rotation group, Langevin. 2000 Math Subject Classification: 62F99, 94C15, 22C05, 05C12, 1
Vector diffusion maps and random matrices with random blocks
, 2014
"... Vector diffusion maps (VDM) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of VDM, i.e the case where there is no structu ..."
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Vector diffusion maps (VDM) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of VDM, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the VDM algorithm- whether there is signal or not. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good. 1
Connection graph Laplacian methods can be made robust to noise,” submitted
, 2014
"... Recently, several data analytic techniques based on connection graph laplacian (CGL) ideas have appeared in the literature. At this point, the properties of these methods are starting to be understood in the setting where the data is observed without noise. We study the impact of additive noise on t ..."
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Recently, several data analytic techniques based on connection graph laplacian (CGL) ideas have appeared in the literature. At this point, the properties of these methods are starting to be understood in the setting where the data is observed without noise. We study the impact of additive noise on these methods, and show that they are remarkably robust. As a by-product of our analysis, we propose modifications of the standard algorithms that increase their robustness to noise. We illustrate our results in numerical simulations. 1
SYNC-RANK: ROBUST RANKING, CONSTRAINED RANKING AND RANK AGGREGATION VIA EIGENVECTOR AND SDP SYNCHRONIZATION
, 2015
"... Abstract. We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), ..."
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Abstract. We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine learning. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose usefulness has been demonstrated in numerous applications in recent years in computer vision and graphics, sensor network localization and structural biology. Its least squares solution can be approximated by either a spectral or a semidefinite programming (SDP) relaxation, followed by a rounding procedure. We show extensive numerical simulations on both synthetic and real-world data sets (Premier League soccer games, a Halo 2 game tournament and NCAA College Basketball games), which show that our proposed method compares favorably to other ranking methods from the recent literature. Existing theoretical guarantees on the group synchronization problem imply lower bounds on the largest amount of noise permissible in the data while still achieving an approximate recovery of the ground truth ranking. We propose a similar synchronization-based algorithm for the rank-aggregation problem, which integrates in a globally consistent ranking many pairwise rank-offsets or partial rankings, given by different rating systems on the same set of items, an approach which yields significantly more accurate results than other aggregation methods, including Rank-Centrality, a recent state-of-the-art algorithm. Furthermore, we discuss the problem of semi-supervised ranking when there is available information on the ground truth rank of a subset of players, and propose an algorithm based on SDP
POINT LOCALIZATION AND DENSITY ESTIMATION FROM ORDINAL KNN GRAPHS USING SYNCHRONIZATION
"... We consider the problem of embedding unweighted, directed k-nearest neighbor graphs in low-dimensional Euclidean space. The k-nearest neighbors of each vertex provide or-dinal information on the distances between points, but not the distances themselves. Relying only on such ordinal in-formation, al ..."
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We consider the problem of embedding unweighted, directed k-nearest neighbor graphs in low-dimensional Euclidean space. The k-nearest neighbors of each vertex provide or-dinal information on the distances between points, but not the distances themselves. Relying only on such ordinal in-formation, along with the low-dimensionality, we recover the coordinates of the points up to arbitrary similarity transfor-mations (rigid transformations and scaling). Furthermore, we also illustrate the possibility of robustly recovering the underlying density via the Total Variation Maximum Penal-ized Likelihood Estimation (TV-MPLE) method. We make existing approaches scalable by using an instance of a local-to-global algorithm based on group synchronization, recently proposed in the literature in the context of sensor network localization, and structural biology, which we augment with a scale synchronization step. We show our approach compares favorably to the recently proposed Local Ordinal Embedding (LOE) algorithm even in the case of smaller sized problems, and also demonstrate its scalability on large graphs. The above divide-and-conquer paradigm can be of independent interest to the machine learning community when tackling geometric embeddings problems. Index Terms — k-nearest-neighbor graphs, ordinal con-straints, graph embeddings, eigenvector synchronization 1.
unknown title
"... Using a distributed SDP approach to solve simulated protein molecular conformation problems Xingyuan Fang and Kim-Chuan Toh Abstract This paper presents various enhancements to the DISCO algorithm (originally introduced by Leung and Toh [18] for anchor-free graph realization in R d) for applications ..."
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Using a distributed SDP approach to solve simulated protein molecular conformation problems Xingyuan Fang and Kim-Chuan Toh Abstract This paper presents various enhancements to the DISCO algorithm (originally introduced by Leung and Toh [18] for anchor-free graph realization in R d) for applications to conformation of protein molecules in R 3. In our enhanced DISCO algorithm for simulated protein molecular conformation problems, we have incorporated distance information derived from chemistry knowledge such as bond lengths and angles to improve the robustness of the algorithm. We also designed heuristics to detect whether a subgroup is well localized and significantly improved the robustness of the stitching process. Tests are performed on molecules taken from the Protein Data Bank. Given only 20 % of the inter-atomic distances less than 6 ˚A that are corrupted by high level of noises (to simulate noisy distance restraints generated from nuclear magnetic resonance experiments), our improved algorithm is
