In this paper, we consider a class of constrained clustering problems of points in @math , where @math could be rather high. A common feature of these problems is that their optimal clusterings no longer have the locality property (due to the additional constraints), which is a key property required by many algorithms for their unconstrained counterparts. To overcome the difficulty caused by the loss of locality, we present in this paper a unified framework, called Peeling-and-Enclosing (PnE) , to iteratively solve two variants of the constrained clustering problems, constrained @math -means clustering ( @math -CMeans) and constrained @math -median clustering ( @math -CMedian). Our framework is based on two standalone geometric techniques, called Simplex Lemma and Weaker Simplex Lemma , for @math -CMeans and @math -CMedian, respectively. The simplex lemma (or weaker simplex lemma) enables us to efficiently approximate the mean (or median) point of an unknown set of points by searching a small-size grid, independent of the dimensionality of the space, in a simplex (or the surrounding region of a simplex), and thus can be used to handle high dimensional data. If @math and @math are fixed numbers, our framework generates, in nearly linear time ( i.e., @math ), @math @math -tuple candidates for the @math mean or median points, and one of them induces a @math -approximation for @math -CMeans or @math -CMedian, where @math is the number of points. Combining this unified framework with a problem-specific selection algorithm (which determines the best @math -tuple candidate), we obtain a @math -approximation for each of the constrained clustering problems. We expect that our technique will be applicable to other constrained clustering problems without locality.
Besides the traditional clustering models, Balcan considered the problem of finding the clustering with small difference from the unknown ground truth @cite_29 @cite_33 .
@cite_29 @cite_33
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The clustering problem is well known in the database literature for its numerous applications in problems such as customer segmentation, classification and trend analysis. Unfortunately, all known algorithms tend to break down in high dimensional spaces because of the inherent sparsity of the points. In such high dimensional spaces not all dimensions may be relevant to a given cluster. One way of handling this is to pick the closely correlated dimensions and find clusters in the corresponding subspace. Traditional feature selection algorithms attempt to achieve this. The weakness of this approach is that in typical high dimensional data mining applications different sets of points may cluster better for different subsets of dimensions. The number of dimensions in each such cluster-specific subspace may also vary. Hence, it may be impossible to find a single small subset of dimensions for all the clusters. We therefore discuss a generalization of the clustering problem, referred to as the projected clustering problem , in which the subsets of dimensions selected are specific to the clusters themselves. We develop an algorithmic framework for solving the projected clustering problem, and test its performance on synthetic data. We consider the following general correlation-clustering problem [N. Bansal, A. Blum, S. Chawla, Correlation clustering, in: Proc. 43rd Annu. IEEE Symp. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250]: given a graph with real nonnegative edge weights and a 〈+〉 〈-〉 edge labelling, partition the vertices into clusters to minimize the total weight of cut 〈+〉 edges and uncut 〈-〉 edges. Thus, 〈+〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈-〉 edges with large weights encourage the endpoints to belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be best possible by the problem definition.Correlation clustering was introduced by [Correlation clustering, in: Proc. 43rd Annu. IEEE Syrup. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250], motivated by both document clustering and agnostic learning. They proved NP-hardness and gave constant-factor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight. We give an O(log n)-approximation algorithm for the general case based on a linear-programming rounding and the "region-growing" technique. We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach. We also give an O(r3)-approximation algorithm for Kr, r-minor-free graphs. On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APX-hard and difficult to approximate better than Θ(log n). We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information- theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs. Although many successful ensemble clustering approaches have been developed in recent years, there are still two limitations to most of the existing approaches. First, they mostly overlook the issue of uncertain links, which may mislead the overall consensus process. Second, they generally lack the ability to incorporate global information to refine the local links. To address these two limitations, in this paper, we propose a novel ensemble clustering approach based on sparse graph representation and probability trajectory analysis. In particular, we present the elite neighbor selection strategy to identify the uncertain links by locally adaptive thresholds and build a sparse graph with a small number of probably reliable links. We argue that a small number of probably reliable links can lead to significantly better consensus results than using all graph links regardless of their reliability. The random walk process driven by a new transition probability matrix is utilized to explore the global information in the graph. We derive a novel and dense similarity measure from the sparse graph by analyzing the probability trajectories of the random walkers, based on which two consensus functions are further proposed. Experimental results on multiple real-world datasets demonstrate the effectiveness and efficiency of our approach.