Clustering is a fundamental task in data analysis, and spectral clustering has been recognized as a promising approach to it. Given a graph describing the relationship between data, spectral clustering explores the underlying cluster structure in two stages. The first stage embeds the nodes of the graph into real space, and the second stage groups the embedded nodes into several clusters. The use of the @math -means method in the grouping stage is currently standard practice. We present a spectral clustering algorithm that uses convex programming in the grouping stage, and study how well it works. The concept behind the algorithm design lies in the following observation. The nodes with the largest degree in each cluster may be found by computing an enclosing ellipsoid for embedded nodes in real space, and the clusters may be identified by using those nodes. We show that the observations are valid, and the algorithm returns clusters to provide the conductance of graph, if the gap assumption, introduced by Peng el al. at COLT 2015, is satisfied. We also give an experimental assessment of the algorithm's performance.
We describe the results of in @cite_20 . Let @math be the clusters of an optimal @math -way partition for the conductance problem. We choose an algorithm for solving the minimization problem of @math shown in ) and suppose that the algorithm has an approximation ratio of @math . Let @math be the output of a spectral clustering algorithm that uses a @math -means method based on the @math -approximation algorithm. They showed the following statement in Theorem 1.2 of @cite_20 . Set @math as and suppose that @math is so large that @math . After suitable renumbering of the output of the algorithm, we have Accordingly, if @math , those can be written as Here, the notation @math denotes the symmetric difference of the sets @math and @math . The results tell us that, if @math is large, the difference between @math and @math is small and the conductance of @math is close to that of @math . also developed a nearly linear time algorithm and examined the performance.
@cite_20
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Let @math be a ground set of @math elements, and let @math be a family of subsets of @math , @math , with a positive cost @math associated with each @math . Consider the following online version of the set cover problem, described as a game between an algorithm and an adversary. An adversary gives elements to the algorithm from @math one by one. Once a new element is given, the algorithm has to cover it by some set of @math containing it. We assume that the elements of @math and the members of @math are known in advance to the algorithm; however, the set @math of elements given by the adversary is not known in advance to the algorithm. (In general, @math may be a strict subset of @math .) The objective is to minimize the total cost of the sets chosen by the algorithm. Let @math denote the family of sets in @math that the algorithm chooses. At the end of the game the adversary also produces (offline) a family of sets @math that covers @math . The performance of the algorithm is the ratio between the cost of @math and the cost of @math . The maximum ratio, taken over all input sequences, is the competitive ratio of the algorithm. We present an @math competitive deterministic algorithm for the problem and establish a nearly matching @math lower bound for all interesting values of @math and @math . The techniques used are motivated by similar techniques developed in computational learning theory for online prediction (e.g., the WINNOW algorithm) together with a novel way of converting a fractional solution into a deterministic online algorithm. For input @math , let @math denote the set of outputs that are the "legal" answers for a computational problem @math . Suppose @math and members of @math are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input @math , support queries by a user to values of specified locations @math in a legal output @math . When more than one legal output @math exists for a given @math , the local computation algorithm should output in a way that is consistent with at least one such @math . Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of @math -wise independent random variables and Beck's analysis in his algorithmic approach to the Lov ' a sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in polylogarithmic time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying @math -SAT formulas. 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. 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.