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Register a new userAchieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions
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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 ErdH{o}s-Renyi. 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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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.
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.
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.
A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barab{a}si-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Comparison of the performance of the algorithms shows it is beneficial to know the arrival times, not just the number, of the children. The probability of correct classification of a vertex is asymptotically determined by the fraction of vertices arriving before it. Two extensions of Algorithm C are given: the first is based on joint likelihood of the children of a fixed set of vertices; it can sometimes be used to seed the message passing algorithm. The second is the message passing algorithm. Simulation results are given.
Quantum error correction (QEC) is an essential element of physical quantum information processing systems. Most QEC efforts focus on extending classical error correction schemes to the quantum regime. The input to a noisy system is embedded in a coded subspace, and error recovery is performed via an operation designed to perfectly correct for a set of errors, presumably a large subset of the physical noise process. In this paper, we examine the choice of recovery operation. Rather than seeking perfect correction on a subset of errors, we seek a recovery operation to maximize the entanglement fidelity for a given input state and noise model. In this way, the recovery operation is optimum for the given encoding and noise process. This optimization is shown to be calculable via a semidefinite program (SDP), a well-established form of convex optimization with efficient algorithms for its solution. The error recovery operation may also be interpreted as a combining operation following a quantum spreading channel, thus providing a quantum analogy to the classical diversity combining operation.
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