text: 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.
  @cite_29: A common approach to clustering data is to view data objects as points in a metric space, and then to optimize a natural distance-based objective such as the k-median, k-means, or min-sum score. For applications such as clustering proteins by function or clustering images by subject, the implicit hope in taking this approach is that the optimal solution for the chosen objective will closely match the desired “target” clustering (e.g., a correct clustering of proteins by function or of images by who is in them). However, most distance-based objectives, including those mentioned here, are NP-hard to optimize. So, this assumption by itself is not sufficient, assuming P ≠ NP, to achieve clusterings of low-error via polynomial time algorithms. In this article, we show that we can bypass this barrier if we slightly extend this assumption to ask that for some small constant c, not only the optimal solution, but also all c-approximations to the optimal solution, differ from the target on at most some e fraction of points—we call this (c,e)-approximation-stability. We show that under this condition, it is possible to efficiently obtain low-error clusterings even if the property holds only for values c for which the objective is known to be NP-hard to approximate. Specifically, for any constant c > 1, (c,e)-approximation-stability of k-median or k-means objectives can be used to efficiently produce a clustering of error O(e) with respect to the target clustering, as can stability of the min-sum objective if the target clusters are sufficiently large. Thus, we can perform nearly as well in terms of agreement with the target clustering as if we could approximate these objectives to this NP-hard value.
  @cite_33: A common approach for solving clustering problems is to design algorithms to approximately optimize various objective functions (e.g., k-means or min-sum) defined in terms of some given pairwise distance or similarity information. However, in many learning motivated clustering applications (such as clustering proteins by function) there is some unknown target clustering; in such cases the pairwise information is merely based on heuristics and the real goal is to achieve low error on the data. In these settings, an arbitrary c-approximation algorithm for some objective would work well only if any c-approximation to that objective is close to the target clustering. In recent work, Balcan et. al [7] have shown how both for the k-means and k-median objectives this property allows one to produce clusterings of low error, even for values c such that getting a c-approximation to these objective functions is provably NP-hard. In this paper we analyze the min-sum objective from this perspective. While [7] also considered the min-sum problem, the results they derived for this objective were substantially weaker. In this work we derive new and more subtle structural properties for min-sum in this context and use these to design efficient algorithms for producing accurate clusterings, both in the transductive and in the inductive case. We also analyze the correlation clustering problem from this perspective, and point out interesting differences between this objective and k-median, k-means, or min-sum objectives.
summary: 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 .