We study the statistical performance of semidefinite programming (SDP) relaxations for clustering under random graph models. Under the @math Synchronization model, Censored Block Model and Stochastic Block Model, we show that SDP achieves an error rate of the form [ [- (1-o(1) ) n I^* ]. ] Here @math is an appropriate multiple of the number of nodes and @math is an information-theoretic measure of the signal-to-noise ratio. We provide matching lower bounds on the Bayes error for each model and therefore demonstrate that the SDP approach is Bayes optimal. As a corollary, our results imply that SDP achieves the optimal exact recovery threshold under each model. Furthermore, we show that SDP is robust: the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary. Our proof is based on a novel primal-dual analysis of SDP under a unified framework for all three models, and the analysis shows that SDP tightly approximates a joint majority voting procedure.
There is a large array of recent results on community detection and graph clustering, in particular, under Z2, CBM and SBM. The readers are referred to the surveys @cite_27 @cite_57 @cite_22 for comprehensive reviews. Without trying to enumerate this body of work, here we restrict attention to those that study sharp performance bounds, with a particular focus on work on the SDP relaxation approach. A more detailed, quantitative comparison with our results is provided in after our main theorems. To begin, we note that existing work has considered several recovery criteria for an estimator @math of @math : means @math is better than random guess, that is, @math ; means @math for a given @math ; means @math @cite_27 .
@cite_57 @cite_27 @cite_22
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We investigate the clustering performances of the relaxed @math means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed @math means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed @math means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed @math means. The stochastic block model (SBM) with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behaviour. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability @math within clusters and @math across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower-bounds on the scaling of @math to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if @math and @math are constant (with @math ), recovering the communities with high probability is possible if @math and impossible if @math . In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst-case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest it may achieve the threshold. An efficient algorithm which succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: @math -means and @math -median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are @math clusters in @math and data from each cluster consists of @math points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these @math clusters as the optimal integral solution? For the @math -median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation @math between the balls. In other words, the pairwise center separation is @math . Under the same distributional model, the @math -means LP relaxation fails to recover such clusters at separation as large as @math . Yet, if we enforce PSD constraints on the @math -means LP, we get exact cluster recovery at center separation @math . In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the @math -means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest. We consider @math -median clustering in finite metric spaces and @math -means clustering in Euclidean spaces, in the setting where @math is part of the input (not a constant). For the @math -means problem, show that if the optimal @math -means clustering of the input is more expensive than the optimal @math -means clustering by a factor of @math , then one can achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math by using a variant of Lloyd's algorithm. In this work we substantially improve this approximation guarantee. We show that given only the condition that the @math -means optimal is more expensive than the @math -means optimal by a factor @math for some constant @math , we can obtain a PTAS. In particular, under this assumption, for any @math we achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math , and exponential in @math and @math . We thus decouple the strength of the assumption from the quality of the approximation ratio. We also give a PTAS for the @math -median problem in finite metrics under the analogous assumption as well. For @math -means, we in addition give a randomized algorithm with improved running time of @math . Our technique also obtains a PTAS under the assumption of that all @math approximations are @math -close to a desired target clustering, in the case that all target clusters have size greater than @math and @math is constant. Note that the motivation of is that for many clustering problems, the objective function is only a proxy for the true goal of getting close to the target. From this perspective, our improvement is that for @math -means in Euclidean spaces we reduce the distance of the clustering found to the target from @math to @math when all target clusters are large, and for @math -median we improve the largeness'' condition needed in the work of to get exactly @math -close from @math to @math . Our results are based on a new notion of clustering stability. New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model @math , @math vertices are split into @math communities of relative size @math , and vertices in community @math and @math connect independently with probability @math . This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function @math , which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of @math ) tends to infinity. We give a lower bound of @math for the degree-4 Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, we show that on an Erd "os-R 'enyi graph @math , with high probability there is a feasible point for the degree-4 SOS relaxation of the clique problem with an objective value of @math , so that the program cannot distinguish between a random graph and a random graph with a planted clique of size @math . This bound is tight. We build on the works of Deshpande and Montanari and , who give lower bounds of @math and @math respectively. We improve on their results by making a perturbation to the SDP solution proposed in their work, then showing that this perturbation remains PSD as the objective value approaches @math . In an independent work, Hopkins, Kothari and Potechin [HKP15] have obtained a similar lower bound for the degree- @math SOS relaxation. The stochastic block model with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behavior. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability @math within clusters and @math across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower bounds on the scaling of @math to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if @math and @math are constant (with @math ), recovering the communities with high probability is possible if @math and is impossible if $( + 2 ) - . In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest that it may achieve the threshold. An efficient algorithm that succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure. We consider the problem of estimating the discrete clustering structures under Sub-Gaussian Mixture Models. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solutions to the SDP are not integer-valued in general, their estimation errors can be upper bounded in terms of the error of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. To the best of our knowledge, this is the first exponentially decaying error bound for convex relaxations of mixture models, and our results reveal the "global-to-local" mechanism that drives the performance of the SDP relaxation. A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact, improving on existing exact recovery results for convex relaxations. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of @math fraction of the points for any @math . As a special case, we show that under the @math -dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as @math , which complements previous exact recovery results that require constant separation. Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, @math -median, and @math -means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. We study clustering problems with outliers. More specifically, we look at Uncapacitated Facility Location (UFL), @math -Median, and @math -Means. In UFL with outliers, we have to open some centres, discard up to @math points of @math and assign every other point to the nearest open centre, minimizing the total assignment cost plus centre opening costs. In @math -Median and @math -Means, we have to open up to @math centres but there are no opening costs. In @math -Means, the cost of assigning @math to @math is @math . We present several results. Our main focus is on cases where @math is a doubling metric or is the shortest path metrics of graphs from a minor-closed family of graphs. For uniform-cost UFL with outliers on such metrics we show that a multiswap simple local search heuristic yields a PTAS. With a bit more work, we extend this to bicriteria approximations for the @math -Median and @math -Means problems in the same metrics where, for any constant @math , we can find a solution using @math centres whose cost is at most a @math -factor of the optimum and uses at most @math outliers. We also show that natural local search heuristics that do not violate the number of clusters and outliers for @math -Median (or @math -Means) will have unbounded gap even in Euclidean metrics. Furthermore, we show how our analysis can be extended to general metrics for @math -Means with outliers to obtain a @math bicriteria. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . Community detection, which aims to cluster @math nodes in a given graph into @math distinct groups based on the observed undirected edges, is an important problem in network data analysis. In this paper, the popular stochastic block model (SBM) is extended to the generalized stochastic block model (GSBM) that allows for adversarial outlier nodes, which are connected with the other nodes in the graph in an arbitrary way. Under this model, we introduce a procedure using convex optimization followed by @math -means algorithm with @math . Both theoretical and numerical properties of the method are analyzed. A theoretical guarantee is given for the procedure to accurately detect the communities with small misclassification rate under the setting where the number of clusters can grow with @math . This theoretical result admits to the best-known result in the literature of computationally feasible community detection in SBM without outliers. Numerical results show that our method is both computationally fast and robust to different kinds of outliers, while some popular computationally fast community detection algorithms, such as spectral clustering applied to adjacency matrices or graph Laplacians, may fail to retrieve the major clusters due to a small portion of outliers. We apply a slight modification of our method to a political blogs data set, showing that our method is competent in practice and comparable to existing computationally feasible methods in the literature. To the best of the authors' knowledge, our result is the first in the literature in terms of clustering communities with fast growing numbers under the GSBM where a portion of arbitrary outlier nodes exist. One of the fundamental clustering problems is to assign @math points into @math clusters based on minimal sum-of-squared distances (MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP) problem. We show that our 0-1 SDP model provides a unified framework for several clustering approaches such as normalized k-cut and spectral clustering. Moreover, the 0-1 SDP model allows us to solve the underlying problem approximately via the linear programming and SDP relaxations. Second, we consider the issue of how to extract a feasible solution of the original 0-1 SDP model from the optimal solution of the relaxed SDP problem. By using principal component analysis, we develop a rounding procedure to construct a feasible partitioning from a solution of the relaxed problem. In our rounding procedure, we need to solve a K-means clustering problem in @math , which can be done in @math time. In case of biclustering, the running time of our rounding procedure can be reduced to @math . We show that our algorithm provides a 2-approximate solution to the original problem. Promising numerical results for biclustering based on our new method are reported. The @math -Means clustering problem on @math points is NP-Hard for any dimension @math , however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an @math time dynamic programming algorithm that uses @math space. It turns out that the problem has been considered under a different name more than twenty years ago. We present all the existing work that had been overlooked and compare the various solutions theoretically. Moreover, we show how to reduce the space usage for some of them, as well as generalize them to data structures that can quickly report an optimal @math -Means clustering for any @math . Finally we also generalize all the algorithms to work for the absolute distance and to work for any Bregman Divergence. We complement our theoretical contributions by experiments that compare the practical performance of the various algorithms. The problem of estimating the phases (angles) of a complex unit-modulus vector @math from their noisy pairwise relative measurements @math , where @math is a complex-valued Gaussian random matrix, is known as phase synchronization. The maximum likelihood estimator (MLE) is a solution to a unit--modulus-constrained quadratic programming problem, which is nonconvex. Existing works have proposed polynomial-time algorithms such as a semidefinite programming (SDP) relaxation or the generalized power method (GPM). Numerical experiments suggest that both of these methods succeed with high probability for @math up to @math , yet existing analyses only confirm this observation for @math up to @math . In this paper, we bridge the gap by proving that the SDP relaxation is tight for @math , and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tight... We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz. In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the @math function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve @math distribution-dependent regret and @math distribution-independent regret, where @math is the time horizon. We apply our results to the @math -MAX problem and expected utility maximization problems. In particular, for @math -MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first @math bound on the @math -approximation regret of its online problem, for any @math . The @math -means clustering algorithm is a ubiquitous tool in data mining and machine learning that shows promising performance. However, its high computational cost has hindered its applications in broad domains. Researchers have successfully addressed these obstacles with dimensionality reduction methods. Recently, [1] develop a state-of-the-art random projection (RP) method for faster @math -means clustering. Their method delivers many improvements over other dimensionality reduction methods. For example, compared to the advanced singular value decomposition based feature extraction approach, [1] reduce the running time by a factor of @math for data matrix @math with @math data points and @math features, while losing only a factor of one in approximation accuracy. Unfortunately, they still require @math for matrix multiplication and this cost will be prohibitive for large values of @math and @math . To break this bottleneck, we carefully build a sparse embedded @math -means clustering algorithm which requires @math ( @math denotes the number of non-zeros in @math ) for fast matrix multiplication. Moreover, our proposed algorithm improves on [1]'s results for approximation accuracy by a factor of one. Our empirical studies corroborate our theoretical findings, and demonstrate that our approach is able to significantly accelerate @math -means clustering, while achieving satisfactory clustering performance. The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math .