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Register a new userJoint Community Detection and Rotational Synchronization via Semidefinite Programming
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In the presence of heterogeneous data, where randomly rotated objects fall into multiple underlying categories, it is challenging to simultaneously classify them into clusters and synchronize them based on pairwise relations. This gives rise to the joint problem of community detection and synchronization. We propose a series of semidefinite relaxations, and prove their exact recovery when extending the celebrated stochastic block model to this new setting where both rotations and cluster identities are to be determined. Numerical experiments demonstrate the efficacy of our proposed algorithms and confirm our theoretical result which indicates a sharp phase transition for exact recovery.
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We propose a new hierarchy of semidefinite programming relaxations for inference problems. As test cases, we consider the problem of community detection in block models. The vertices are partitioned into $k$ communities, and a graph is sampled conditional on a prescribed number of inter- and intra-community edges. The problem of detection, where we are to decide with high probability whether a graph was drawn from this model or the uniform distribution on regular graphs, is conjectured to undergo a computational phase transition at a point called the Kesten-Stigum (KS) threshold. In this work, we consider two models of random graphs namely the well-studied (irregular) stochastic block model and a distribution over random regular graphs well call the Degree Regular Block Model. For both these models, we show that sufficiently high constant levels of our hierarchy can perform detection arbitrarily close to the KS threshold and that our algorithm is robust to up to a linear number of adversarial edge perturbations. Furthermore, in the case of Degree Regular Block Model (DRBM), we show that below the Kesten-Stigum threshold no constant level can do so. In the case of the (irregular) Stochastic Block Model, it is known that efficient algorithms exist all the way down to this threshold, although none are robust to a linear number of adversarial perturbations of the graph when the average degree is small. More importantly, there is little complexity-theoretic evidence that detection is hard below the threshold. In the DRBM with more than two groups, it has not to our knowledge been proven that any algorithm succeeds down to the KS threshold, let alone that one can do so robustly, and there is a similar dearth of evidence for hardness below this point.
The community detection problem requires to cluster the nodes of a network into a small number of well-connected communities. There has been substantial recent progress in characterizing the fundamental statistical limits of community detection under simple stochastic block models. However, in real-world applications, the network structure is typically dynamic, with nodes that join over time. In this setting, we would like a detection algorithm to perform only a limited number of updates at each node arrival. While standard voting approaches satisfy this constraint, it is unclear whether they exploit the network information optimally. We introduce a simple model for networks growing over time which we refer to as streaming stochastic block model (StSBM). Within this model, we prove that voting algorithms have fundamental limitations. We also develop a streaming belief-propagation (StreamBP) approach, for which we prove optimality in certain regimes. We validate our theoretical findings on synthetic and real data.
We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} sim P$ if $i, j$ are both in the community and $A_{ij} sim Q$ otherwise, for two known probability distributions $P$ and $Q$. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case ($P={{rm Bern}}(p)$ and $Q={{rm Bern}}(q)$ with $p>q$) and the Gaussian case ($P=mathcal{N}(mu,1)$ and $Q=mathcal{N}(0,1)$ with $mu>0$), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If $K=omega( n /log n)$, SDP attains the information-theoretic recovery limits with sharp constants; (2) If $K=Theta(n/log n)$, SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If $K=o(n/log n)$ and $K to infty$, SDP is order-wise suboptimal. The same critical scaling for $K$ is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of $n$ vertices partitioned into multiple communities of equal size $K$. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.
Based on the classical Degree Corrected Stochastic Blockmodel (DCSBM) model for network community detection problem, we propose two novel approaches: principal component clustering (PCC) and normalized principal component clustering (NPCC). Without any parameters to be estimated, the PCC method is simple to be implemented. Under mild conditions, we show that PCC yields consistent community detection. NPCC is designed based on the combination of the PCC and the RSC method (Qin & Rohe 2013). Population analysis for NPCC shows that NPCC returns perfect clustering for the ideal case under DCSBM. PCC and NPCC is illustrated through synthetic and real-world datasets. Numerical results show that NPCC provides a significant improvement compare with PCC and RSC. Moreover, NPCC inherits nice properties of PCC and RSC such that NPCC is insensitive to the number of eigenvectors to be clustered and the choosing of the tuning parameter. When dealing with two weak signal networks Simmons and Caltech, by considering one more eigenvectors for clustering, we provide two refinements PCC+ and NPCC+ of PCC and NPCC, respectively. Both two refinements algorithms provide improvement performances compared with their original algorithms. Especially, NPCC+ provides satisfactory performances on Simmons and Caltech, with error rates of 121/1137 and 96/590, respectively.
Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{Theta(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $dge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.
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