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238
Clustering Partially Observed Graphs via Convex Optimization
"... This paper considers the problem of clustering a partially observed unweighted graph – i.e. one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nod ..."
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Cited by 42 (12 self)
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This paper considers the problem of clustering a partially observed unweighted graph – i.e. one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of ”disagreements ”i.e. the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) lowrank matrix and an (unknown) sparse matrix from their partially observed sum. We show that our algorithm succeeds under certain natural assumptions on the optimal clustering and its disagreements. Our results significantly strengthen existing matrix splitting results for the special case of our clustering problem. Our results directly enhance solutions to the problem of Correlation Clustering (Bansal et al., 2002) with partial observations.
IMPROVED ANALYSIS OF THE SUBSAMPLED RANDOMIZED HADAMARD TRANSFORM
"... Abstract. This paper presents an improved analysis of a structured dimensionreduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous appro ..."
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Cited by 41 (1 self)
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Abstract. This paper presents an improved analysis of a structured dimensionreduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous approaches, and it offers—for the first time—optimal constants in the estimate on the number of dimensions required for the embedding. 1.
Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model
"... In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model ..."
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Cited by 41 (0 self)
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In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model, a variant of the classical planted partition model. The standard approach to spectral clustering of graphs is to compute the bottom k singular vectors or eigenvectors of a suitable graph Laplacian, project the nodes of the graph onto these vectors, and then use an iterative clustering algorithm on the projected nodes. However a challenge with applying this approach to graphs generated from the EPP model is that unnormalized Laplacians do not work, and normalized Laplacians do not concentrate well when the graph has a number of low degree nodes. We resolve this issue by introducing the notion of a degreecorrected graph Laplacian. For graphs with many low degree nodes, degree correction has a regularizing effect on the Laplacian. Our spectral clustering algorithm projects the nodes in the graph onto the bottom k right singular vectors of the degreecorrected randomwalk Laplacian, and clusters the nodes in this subspace. We show guarantees on the performance of this algorithm, demonstrating that it outputs the correct partition under a wide range of parameter values. Unlike some previous work, our algorithm does not require access to any generative parameters of the model.
A Tensor Spectral Approach to Learning Mixed Membership Community Models
"... Detecting hidden communities from observed interactions is a classical problem. Theoretical analysis of community detection has so far been mostly limited to models with nonoverlapping communities such as the stochastic block model. In this paper, we provide guaranteed community detection for a fam ..."
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Cited by 26 (4 self)
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Detecting hidden communities from observed interactions is a classical problem. Theoretical analysis of community detection has so far been mostly limited to models with nonoverlapping communities such as the stochastic block model. In this paper, we provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced in Airoldi et al. (2008). This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning communities in these models via a tensor spectral decomposition approach. Our estimator uses loworder moment tensor of the observed network, consisting of 3star counts. Our learning method is based on simple linear algebraic operations such as singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters, and present a careful finite sample analysis of our learning method. Additionally, our results match the best known scaling requirements for the special case of the (homogeneous) stochastic block model.
Compressed Sensing Off the Grid
, 2012
"... This work investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized ..."
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Cited by 25 (1 self)
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This work investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(s log s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Numerical experiments are performed to illustrate the effectiveness of the proposed method.
Recursive robust pca or recursive sparse recovery in large but structured noise
 in IEEE Intl. Symp. on Information Theory (ISIT
, 2013
"... This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more informati ..."
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Cited by 22 (17 self)
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This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please contact
Paved with good intentions: Analysis of a randomized Kaczmarz method
, 2012
"... ABSTRACT. The block Kaczmarz method is an iterative scheme for solving overdetermined leastsquares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized contro ..."
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Cited by 21 (6 self)
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ABSTRACT. The block Kaczmarz method is an iterative scheme for solving overdetermined leastsquares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving of the matrix, a partition of the rows into wellconditioned blocks. The operator theory literature provides detailed information about the existence and construction of good row pavings. Together, these results yield an efficient block Kaczmarz scheme that applies to many overdetermined leastsquares problem. 1.
Phase Retrieval from Coded Diffraction Patterns
, 2013
"... This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the inten ..."
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Cited by 21 (5 self)
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This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the intensity of its diffraction pattern, each modulation thereby producing a sort of coded diffraction pattern. We show that PhaseLift, a recent convex programming technique, recovers the phase information exactly from a number of random modulations, which is polylogarithmic in the number of unknowns. Numerical experiments with noiseless and noisy data complement our theoretical analysis and illustrate our approach.