The Highest Dimensional Stochastic Blockmodel with a Regularized Estimator

Rohe, Karl; Qin, Tai; Fan, Haoyang

doi:10.48550/arxiv.1206.2380

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…First, although our recovery guarantees match those previously identified in the literature, these bounds may not be the best possible. Rohe et al [46] recently established that an N -node random graph sampled from the Stochastic Blockmodel can be partitioned into dense subgraphs of size Ω(log 4 N ) using a regularized maximum likelihood estimator. It is unclear if such a bound can be attained for our relaxation.…”

Section: Resultsmentioning

confidence: 99%

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

Ames

2015

J Optim Theory Appl

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We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and 1 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our convex relaxation for certain program inputs. In particular, we establish exact recovery in the case that the input graph contains a single planted clique plus noise in the form of corrupted adjacency relationships. We consider two constructions for this noise. In the first, noise is introduced by an adversary deterministically deleting edges within the planted clique and placing diversionary edges. In the second, these edge corruptions are performed at random. Analogous recovery guarantees for identifying the densest subgraph of fixed size in a bipartite graph are also established, and results of numerical simulations for randomly generated graphs are included to demonstrate the efficacy of our algorithm.

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Section: Resultsmentioning

confidence: 99%

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

Ames

2015

J Optim Theory Appl

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“…However, all in all, I am not aware of any estimator for the stochastic blockmodel that works whenever the number of blocks is small compared to the number of nodes. The best result till date is in the very recent manuscript of Rohe et al [86], who prove that a penalized likelihood estimator works whenever k is comparable to n "up to log factors." The following theorem says that the USVT estimator M gives a complete solution to the estimation problem in the stochastic blockmodel if k ≪ n, with no further conditions required.…”

Section: The Stochastic Blockmodelmentioning

confidence: 99%

Matrix estimation by Universal Singular Value Thresholding

Chatterjee¹

2015

Ann. Statist.

376

538

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Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Cand\`{e}s and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure." Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation and generalized Bradley--Terry models for pairwise comparison.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1272 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…. , K. For example, the submodel may restrict H so that H(Z i , Z j ) depends only on whether Z i = Z j or not; this assumption could reflect homogeneity of the classes, and is explored in Rohe, Qin and Fan (2012). This submodel is identifiable under ordering of π, and the latent structure might be more gracefully described as a partition, that is, a variable X ∈ {0, 1} n×n satisfying X(i, j) = 1 iff Z i = Z j .…”

Section: Preliminariesmentioning

confidence: 99%

Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels

Bickel¹,

Choi²,

Chang³

et al. 2013

Ann. Statist.

177

229

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Variational methods for parameter estimation are an active research area, potentially offering computationally tractable heuristics with theoretical performance bounds. We build on recent work that applies such methods to network data, and establish asymptotic normality rates for parameter estimates of stochastic blockmodel data, by either maximum likelihood or variational estimation. The result also applies to various sub-models of the stochastic blockmodel found in the literature.

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The Highest Dimensional Stochastic Blockmodel with a Regularized Estimator

Cited by 4 publications

References 17 publications

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

Matrix estimation by Universal Singular Value Thresholding

Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels

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