A basic problem in spectral clustering is the following. If a solution obtained from the spectral relaxation is close to an integral solution, is it possible to find this integral solution even though they might be in completely different basis? In this paper, we propose a new spectral clustering algorithm. It can recover a @math -partition such that the subspace corresponding to the span of its indicator vectors is @math close to the original subspace in spectral norm with @math being the minimum possible ( @math always). Moreover our algorithm does not impose any restriction on the cluster sizes. Previously, no algorithm was known which could find a @math -partition closer than @math . We present two applications for our algorithm. First one finds a disjoint union of bounded degree expanders which approximate a given graph in spectral norm. The second one is for approximating the sparsest @math -partition in a graph where each cluster have expansion at most @math provided @math where @math is the @math eigenvalue of Laplacian matrix. This significantly improves upon the previous algorithms, which required @math .
One of the first spectral clustering algorithms with worst case guarantees was given in @cite_12 for the graph partitioning problem assuming certain conditions on the internal versus external conductance. The problem of finding a @math -partition so as to minimize the spectral norm was first introduced by @cite_4 in the context of learning mixtures of Gaussians. The best known approximation factor is @math due to @cite_6 .
@cite_4 @cite_6 @cite_12
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Aiming to unify known results about clustering mixtures of distributions under separation conditions, Kumar and Kannan[2010] introduced a deterministic condition for clustering datasets. They showed that this single deterministic condition encompasses many previously studied clustering assumptions. More specifically, their proximity condition requires that in the target @math -clustering, the projection of a point @math onto the line joining its cluster center @math and some other center @math , is a large additive factor closer to @math than to @math . This additive factor can be roughly described as @math times the spectral norm of the matrix representing the differences between the given (known) dataset and the means of the (unknown) target clustering. Clearly, the proximity condition implies center separation -- the distance between any two centers must be as large as the above mentioned bound. In this paper we improve upon the work of Kumar and Kannan along several axes. First, we weaken the center separation bound by a factor of @math , and secondly we weaken the proximity condition by a factor of @math . Using these weaker bounds we still achieve the same guarantees when all points satisfy the proximity condition. We also achieve better guarantees when only @math -fraction of the points satisfy the weaker proximity condition. The bulk of our analysis relies only on center separation under which one can produce a clustering which (i) has low error, (ii) has low @math -means cost, and (iii) has centers very close to the target centers. Our improved separation condition allows us to match the results of the Planted Partition Model of McSherry[2001], improve upon the results of [2006], and improve separation results for mixture of Gaussian models in a particular setting. Let (G) be the minimum conductance of an undirected graph G, and let 0= = 2, (G) = O(k) , and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if $ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut. In this paper we study variants of the widely used spectral clustering that partitions a graph into @math clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix and (2) grouping the embedded points into @math clusters via @math -means algorithms. We show that, for a wide class of graphs, spectral clustering gives a good approximation of the optimal clustering. While this approach was proposed in the early 1990s and has comprehensive applications, prior to our work similar results were known only for graphs generated from stochastic models. We also give a nearly linear time algorithm for partitioning well-clustered graphs based on computing a matrix exponential and approximate nearest neighbor data structures. We describe a new approximation algorithm for Max Cut. Our algorithm runs in @math time, where @math is the number of vertices, and achieves an approximation ratio of @math . In instances in which an optimal solution cuts a @math fraction of edges, our algorithm finds a solution that cuts a @math fraction of edges. Our main result is a variant of spectral partitioning, which can be implemented in nearly linear time. Given a graph in which the Max Cut optimum is a @math fraction of edges, our spectral partitioning algorithm finds a set @math of vertices and a bipartition @math of @math such that at least a @math fraction of the edges incident on @math have one endpoint in @math and one endpoint in @math . (This can be seen as an analogue of Cheeger's inequality for the smallest eigenvalue of the adjacency matrix of a graph.) Iterating this procedure yields the approximation results stated above. A different, more complicated, var... 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 . Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral graph partitioning under the stochastic block model are well known. In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model. We derive an error bound for a spectral hypergraph partitioning algorithm under this model using matrix concentration inequalities. To the best of our knowledge, this is the first consistency result related to partitioning non-uniform hypergraphs. The min-sum k -clustering problem is to partition a metric space (P,d) into k clusters C 1,…,C k ⊆P such that @math is minimized. We show the first efficient construction of a coreset for this problem. Our coreset construction is based on a new adaptive sampling algorithm. With our construction of coresets we obtain two main algorithmic results. The first result is a sublinear-time (4+e)-approximation algorithm for the min-sum k-clustering problem in metric spaces. The running time of this algorithm is @math for any constant k and e, and it is o(n 2) for all k=o(log n log log n). Since the full description size of the input is Θ(n 2), this is sublinear in the input size. The fastest previously known o(log n)-factor approximation algorithm for k>2 achieved a running time of Ω(n k ), and no non-trivial o(n 2)-time algorithm was known before. Our second result is the first pass-efficient data streaming algorithm for min-sum k-clustering in the distance oracle model, i.e., an algorithm that uses poly(log n,k) space and makes 2 passes over the input point set, which arrives in form of a data stream in arbitrary order. It computes an implicit representation of a clustering of (P,d) with cost at most a constant factor larger than that of an optimal partition. Using one further pass, we can assign each point to its corresponding cluster. To develop the coresets, we introduce the concept of α -preserving metric embeddings. Such an embedding satisfies properties that the distance between any pair of points does not decrease and the cost of an optimal solution for the considered problem on input (P,d′) is within a constant factor of the optimal solution on input (P,d). In other words, the goal is to find a metric embedding into a (structurally simpler) metric space that approximates the original metric up to a factor of α with respect to a given problem. We believe that this concept is an interesting generalization of coresets. A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, @math the minimum degree, and @math the maximum degree. We show that every graph has a domatic partition with @math dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a @math -approximation algorithm for domatic number, since the domatic number is always at most @math . We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every @math , a @math -approximation implies that @math . This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with @math dominating sets, where the "o(1)" term goes to zero as @math increases. This can be turned into an efficient algorithm that produces a domatic partition of @math sets. Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any @math @math -dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses @math samples and runs in time @math , both significantly lower than previously known. The constant factor @math is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that @math samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for one-dimensional mixtures that uses @math samples and runs in time @math . Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the @math distance to an unknown underlying distribution. Spectral clustering is a popular and successful approach for partitioning the nodes of a graph into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. In order to partition into k clusters, one first computes an approximation of the bottom k eigenvectors of the (normalized) Laplacian of G, uses it to embed the vertices of G into k-dimensional Euclidean space R^k, and then partitions the resulting points via a k-means clustering algorithm. It is an important task for theory to explain the success of spectral clustering. (COLT, 2015) made an important step in this direction. They showed that spectral clustering provably works if the gap between the (k+1)-th and the k-th eigenvalue of the normalized Laplacian is sufficiently large. They proved a structural and an algorithmic result. The algorithmic result needs a considerably stronger gap assumption and does not analyze the standard spectral clustering paradigm; it replaces spectral embedding by heat kernel embedding and k-means clustering by locality sensitive hashing. We extend their work in two directions. Structurally, we improve the quality guarantee for spectral clustering by a factor of k and simultaneously weaken the gap assumption. Algorithmically, we show that the standard paradigm for spectral clustering works. Moreover, it even works with the same gap assumption as required for the structural result. We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were @math for both directed and undirected graphs, and @math for undirected graphs with @math , where @math is the number of nodes in the input graph. Our first algorithm has approximation ratio @math , which is @math except for very large values of @math , namely, @math . This algorithm is based on a new result on @math -connected @math -critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio @math for all values of @math . Combining these two gives an algorithm with approximation ratio @math , which asymptotically improves the best known approximation guarantee for directed graphs for all values of @math , and for undirected graphs for @math . Moreover, this is the first algorithm that has an approximation guarantee better than @math for all values of @math . Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation. The Euclidean @math -means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of @math points in Euclidean space @math , and the goal is to choose @math centers in @math so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general @math and a @math -approximation which runs in time @math . At the other extreme, the only known computational complexity result for this problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in @math can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all @math . In this paper we provide the first hardness of approximation for the Euclidean @math -means problem. Concretely, we show that there exists a constant @math such that it is NP-hard to approximate the @math -means objective to within a factor of @math . We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform @math , a known hard vertex cover instance, by taking a graph product with a suitably chosen graph @math , and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest. 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. We consider the problem of selecting the best subset of exactly @math columns from an @math matrix @math . We present and analyze a novel two-stage algorithm that runs in @math time and returns as output an @math matrix @math consisting of exactly @math columns of @math . In the first (randomized) stage, the algorithm randomly selects @math columns according to a judiciously-chosen probability distribution that depends on information in the top- @math right singular subspace of @math . In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly @math columns from the set of columns selected in the first stage. Let @math be the @math matrix containing those @math columns, let @math denote the projection matrix onto the span of those columns, and let @math denote the best rank- @math approximation to the matrix @math . Then, we prove that, with probability at least 0.8, @math This Frobenius norm bound is only a factor of @math worse than the best previously existing existential result and is roughly @math better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, @math This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on @math , whereas previous results depend on @math ; if these two quantities are comparable, then our bound is asymptotically worse by a @math factor. We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph @math and a root node @math , we seek a single spanning tree @math of minimum stretch , where the stretch of @math is defined to be the maximum ratio, over all terminal sets @math , of the cost of the minimal sub-tree @math of @math that connects @math to @math to the cost of an optimal Steiner tree connecting @math to @math in @math . Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. We provide a polynomial time UST construction for general graphs with @math -stretch. We also give a polynomial time polylogarithmic-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first sub polynomial-stretch ( @math for any @math ) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs. The min-sum k-clustering problem in a metric space is to find a partition of the space into k clusters as to minimize the total sum of distances between pairs of points assigned to the same cluster. We give the first polynomial time non-trivial approximation algorithm for this problem. The algorithm provides an @math approximation to the min-sum k -clustering problem in general metric spaces, with running time @math . The result is based on embedding of metric spaces into hierarchically separated trees. We also provide a bicriteria approximation result that provides a constant approximation factor solution with only a constant factor increase in the number of clusters. This result is obtained by modifying and drawing ideas from recently developed primal dual approximation algorithms for facility location. Guruswami and Sinop give a @math approximation guarantee for the non-uniform Sparsest Cut problem by solving @math -level Lasserre semidefinite constraints, provided that the generalized eigenvalues of the Laplacians of the cost and demand graphs satisfy a certain spectral condition, namely, @math . Their key idea is a rounding technique that first maps a vector-valued solution to @math using appropriately scaled projections onto Lasserre vectors. In this paper, we show that similar projections and analysis can be obtained using only @math triangle inequality constraints. This results in a @math approximation guarantee for the non-uniform Sparsest Cut problem by adding only @math triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition, @math , holds. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are @math observed contexts and @math arms of the bandit. The observed context influences the reward obtained through a latent confounder variable with cardinality @math ( @math ). The arm choice and the latent confounder causally determines the reward while the observed context is correlated with the confounder. Under this model, the @math mean reward matrix @math (for each context in @math and each arm in @math ) factorizes into non-negative factors @math ( @math ) and @math ( @math ). This insight enables us to propose an @math -greedy NMF-Bandit algorithm that designs a sequence of interventions (selecting specific arms), that achieves a balance between learning this low-dimensional structure and selecting the best arm to minimize regret. Our algorithm achieves a regret of @math at time @math , as compared to @math for conventional contextual bandits, assuming a constant gap between the best arm and the rest for each context. These guarantees are obtained under mild sufficiency conditions on the factors that are weaker versions of the well-known Statistical RIP condition. We further propose a class of generative models that satisfy our sufficient conditions, and derive a lower bound of @math . These are the first regret guarantees for online matrix completion with bandit feedback, when the rank is greater than one. We further compare the performance of our algorithm with the state of the art, on synthetic and real world data-sets. 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.