Clustering is an effective technique in data mining to generate groups that are the matter of interest. Among various clustering approaches, the family of k-means algorithms and min-cut algorithms gain most popularity due to their simplicity and efficacy. The classical k-means algorithm partitions a number of data points into several subsets by iteratively updating the clustering centers and the associated data points. By contrast, a weighted undirected graph is constructed in min-cut algorithms which partition the vertices of the graph into two sets. However, existing clustering algorithms tend to cluster minority of data points into a subset, which shall be avoided when the target dataset is balanced. To achieve more accurate clustering for balanced dataset, we propose to leverage exclusive lasso on k-means and min-cut to regulate the balance degree of the clustering results. By optimizing our objective functions that build atop the exclusive lasso, we can make the clustering result as much balanced as possible. Extensive experiments on several large-scale datasets validate the advantage of the proposed algorithms compared to the state-of-the-art clustering algorithms.
In the literature, the classical @math -means and its variants have been applied to many data mining applications. For example, Mehrdad @cite_11 propose a harmony @math -means (HKM) algorithm based on harmony search optimization method and applied it to document clustering. HKM can be proved by means of finite Markov chain theory to converge to the global optimum. Zhang @cite_2 propose a new neighborhood density method for selecting initial cluster centers for @math -means clustering. Deepak @cite_3 employ quantization schemes to retain the outcome of clustering operations. Although these methods get good performance, they have not considered how to achieve balanced clustering result when the given data points are evenly distributed. By contrast, we aim to develop a balanced @math -means clustering algorithm that well addresses this issue.
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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. 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. There has been much progress on efficient algorithms for clustering data points generated by a mixture of @math probability distributions under the assumption that the means of the distributions are well-separated, i.e., the distance between the means of any two distributions is at least @math standard deviations. These results generally make heavy use of the generative model and particular properties of the distributions. In this paper, we show that a simple clustering algorithm works without assuming any generative (probabilistic) model. Our only assumption is what we call a "proximity condition": the projection of any data point onto the line joining its cluster center to any other cluster center is @math standard deviations closer to its own center than the other center. Here the notion of standard deviations is based on the spectral norm of the matrix whose rows represent the difference between a point and the mean of the cluster to which it belongs. We show that in the generative models studied, our proximity condition is satisfied and so we are able to derive most known results for generative models as corollaries of our main result. We also prove some new results for generative models - e.g., we can cluster all but a small fraction of points only assuming a bound on the variance. Our algorithm relies on the well known @math -means algorithm, and along the way, we prove a result of independent interest -- that the @math -means algorithm converges to the "true centers" even in the presence of spurious points provided the initial (estimated) centers are close enough to the corresponding actual centers and all but a small fraction of the points satisfy the proximity condition. Finally, we present a new technique for boosting the ratio of inter-center separation to standard deviation.