Sharp performance bounds for graph clustering via convex optimization

Vinayak, Ramya Korlakai; Oymak, Samet; Hassibi, Babak

doi:10.1109/icassp.2014.6855219

Cited by 17 publications

(16 citation statements)

References 30 publications

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“…where (41) is due to (39), (42) is by definition of η, and (43) follows from Lemma 13. Similarly, for i ∈ C 2 , A ij σ i σ j is stochastically larger than X − R − 1, where X ∼ Binom(n − K n , a log n n ) and R ∼ Binom(K n , b log n n ).…”

Section: Applying the Union Bound Yieldsmentioning

confidence: 99%

Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions

Hajek

2016

IEEE Trans. Inform. Theory

118

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Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model of two equal-sized clusters. The same was shown for the case of a single cluster and outliers. Extending the proof techniques, in this paper it is shown that SDP relaxations also achieve the sharp recovery threshold in the following cases: (1) Binary stochastic block model with two clusters of sizes proportional to network size but not necessarily equal; (2) Stochastic block model with a fixed number of equal-sized clusters; (3) Binary censored block model with the background graph being Erdős-Rényi. Furthermore, a sufficient condition is given for an SDP procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers. These results demonstrate the versatility of SDP relaxation as a simple, general purpose, computationally feasible methodology for community detection.

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Section: Applying the Union Bound Yieldsmentioning

confidence: 99%

Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions

Hajek

2016

IEEE Trans. Inform. Theory

118

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“…In spite of the inherent intractability of clustering, many recent analyses have established that if data is sampled from some distribution of clusterable data, then one can efficiently recover the underlying cluster structure using a variety of clustering algorithms. In particular, the recent results of Abbe et al (2016); Ailon et al (2013); Ames (2014); Ames and Vavasis (2014); Amini and Levina (2018); Cai and Li (2015); Chen and Xu (2014); Chen et al (2014a,b); Guédon and Vershynin (2015); Hajek et al (2015); Lei and Rinaldo (2015); Mathieu and Schudy (2010); Nellore and Ward (2015); Oymak and Hassibi (2011); Qin and Rohe (2013); Rohe et al (2011); Vinayak et al (2014) all establish sufficient conditions under which we can expect to identify the latent cluster structure efficiently. Most of these results assume that the similarity structure of the data can be modeled as a graph sampled from some generalization of the stochastic block model proposed by Holland et al (1983).…”

Section: Introductionmentioning

confidence: 81%

Exact Clustering of Weighted Graphs via Semidefinite Programming

Pirinen,

Ames

2016

Preprint

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As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of nodes not belonging to any cluster. Further, we establish that if noise is weakly obscuring these clusters, i.e, the between-cluster edges are assigned very small weights, then we can recover significantly smaller clusters. For example, we show that in approximately sparse graphs, where the between-cluster weights tend to zero as the size n of the graph tends to infinity, we can recover clusters of size polylogarithmic in n under certain conditions on the distribution of edge weights. Empirical evidence from numerical simulations is also provided to support these theoretical phase transitions to perfect recovery of the cluster structure.

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“…The SDP machinery has also been applied to recover clusters with partially observed graphs [11], [23] and binary matrices [33]. In the converse direction, necessary conditions for the success of particular SDPs are obtained in [32], [13]. In contrast to the previous work mentioned above where the constants are often loose, the recent line of work initiated by [2], [1], and followed by [20], [9], [4], [31] and the current paper, focus on establishing necessary and sufficient conditions in the special case of a fixed number of clusters with sharp constants, attained via SDP relaxations.…”

Section: B Further Literature On Sdp For Cluster Recoverymentioning

confidence: 99%