k-medoids algorithm is a partitional, centroid-based clustering algorithm which uses pairwise distances of data points and tries to directly decompose the dataset with @math points into a set of @math disjoint clusters. However, k-medoids itself requires all distances between data points that are not so easy to get in many applications. In this paper, we introduce a new method which requires only a small proportion of the whole set of distances and makes an effort to estimate an upper-bound for unknown distances using the inquired ones. This algorithm makes use of the triangle inequality to calculate an upper-bound estimation of the unknown distances. Our method is built upon a recursive approach to cluster objects and to choose some points actively from each bunch of data and acquire the distances between these prominent points from oracle. Experimental results show that the proposed method using only a small subset of the distances can find proper clustering on many real-world and synthetic datasets.
 The technique of spectral clustering is widely used to segment a range of data from graphs to images. Our work marks a natural progression of spectral clustering from the original passive unsupervised formulation to our active semi-supervised formulation. We follow the widely used area of constrained clustering and allow supervision in the form of pair wise relations between two nodes: Must-Link and Cannot-Link. Unlike most previous constrained clustering work, our constraints are specified incrementally by querying an oracle (domain expert). Since in practice, each query comes with a cost, our goal is to maximally improve the result with as few queries as possible. The advantages of our approach include: 1) it is principled by querying the constraints which maximally reduce the expected error, 2) it can incorporate both hard and soft constraints which are prevalent in practice. We empirically show that our method significantly outperforms the baseline approach, namely constrained spectral clustering with randomly selected constraints, on UCI benchmark data sets. Semi-supervised clustering seeks to augment traditional clustering methods by incorporating side information provided via human expertise in order to increase the semantic meaningfulness of the resulting clusters. However, most current methods are in the sense that the side information is provided beforehand and selected randomly. This may require a large number of constraints, some of which could be redundant, unnecessary, or even detrimental to the clustering results. Thus in order to scale such semi-supervised algorithms to larger problems it is desirable to pursue an clustering method---i.e. an algorithm that maximizes the effectiveness of the available human labor by only requesting human input where it will have the greatest impact. Here, we propose a novel online framework for active semi-supervised spectral clustering that selects pairwise constraints as clustering proceeds, based on the principle of uncertainty reduction. Using a first-order Taylor expansion, we decompose the expected uncertainty reduction problem into a gradient and a step-scale, computed via an application of matrix perturbation theory and cluster-assignment entropy, respectively. The resulting model is used to estimate the uncertainty reduction potential of each sample in the dataset. We then present the human user with pairwise queries with respect to only the best candidate sample. We evaluate our method using three different image datasets (faces, leaves and dogs), a set of common UCI machine learning datasets and a gene dataset. The results validate our decomposition formulation and show that our method is consistently superior to existing state-of-the-art techniques, as well as being robust to noise and to unknown numbers of clusters. Spectral clustering is a modern and well known method for performing data clustering. However, it depends on the availability of a similarity matrix, which in many applications can be non-trivial to obtain. In this paper, we focus on the problem of performing spectral clustering under a budget constraint, where there is a limit on the number of entries which can be queried from the similarity matrix. We propose two algorithms for this problem, and study them theoretically and experimentally. These algorithms allow a tradeo between computational eciency and actual performance, and are also relevant for the problem of speeding up standard spectral clustering. Semi-supervised clustering uses a small amount of supervised data to aid unsupervised learning. One typical approach specifies a limited number of must-link and cannotlink constraints between pairs of examples. This paper presents a pairwise constrained clustering framework and a new method for actively selecting informative pairwise constraints to get improved clustering performance. The clustering and active learning methods are both easily scalable to large datasets, and can handle very high dimensional data. Experimental and theoretical results confirm that this active querying of pairwise constraints significantly improves the accuracy of clustering when given a relatively small amount of supervision. Given a point set S and an unknown metric d on S, we study the problem of efficiently partitioning S into k clusters while querying few distances between the points. In our model we assume that we have access to one versus all queries that given a point s ∈ S return the distances between s and all other points. We show that given a natural assumption about the structure of the instance, we can efficiently find an accurate clustering using only O(k) distance queries. Our algorithm uses an active selection strategy to choose a small set of points that we call landmarks, and considers only the distances between landmarks and other points to produce a clustering. We use our procedure to cluster proteins by sequence similarity. This setting nicely fits our model because we can use a fast sequence database search program to query a sequence against an entire data set. We conduct an empirical study that shows that even though we query a small fraction of the distances between the points, we produce clusterings that are close to a desired clustering given by manual classification.  Spectral clustering is a widely used method for organizing data that only relies on pairwise similarity measurements. This makes its application to non-vectorial data straight-forward in principle, as long as all pairwise similarities are available. However, in recent years, numerous examples have emerged in which the cost of assessing similarities is substantial or prohibitive. We propose an active learning algorithm for spectral clustering that incrementally measures only those similarities that are most likely to remove uncertainty in an intermediate clustering solution. In many applications, similarities are not only costly to compute, but also noisy. We extend our algorithm to maintain running estimates of the true similarities, as well as estimates of their accuracy. Using this information, the algorithm updates only those estimates which are relatively inaccurate and whose update would most likely remove clustering uncertainty. We compare our methods on several datasets, including a realistic example where similarities are expensive and noisy. The results show a significant improvement in performance compared to the alternatives.  In this article, we address the problem of automatic constraint selection to improve the performance of constraint-based clustering algorithms. To this aim we propose a novel active learning algorithm that relies on a k-nearest neighbors graph and a new constraint utility function to generate queries to the human expert. This mechanism is paired with propagation and refinement processes that limit the number of constraint candidates and introduce a minimal diversity in the proposed constraints. Existing constraint selection heuristics are based on a random selection or on a min-max criterion and thus are either inefficient or more adapted to spherical clusters. Contrary to these approaches, our method is designed to be beneficial for all constraint-based clustering algorithms. Comparative experiments conducted on real datasets and with two distinct representative constraint-based clustering algorithms show that our approach significantly improves clustering quality while minimizing the number of human expert solicitations. Advances in sensing technologies and the growth of the internet have resulted in an explosion in the size of modern datasets, while storage and processing power continue to lag behind. This motivates the need for algorithms that are efficient, both in terms of the number of measurements needed and running time. To combat the challenges associated with large datasets, we propose a general framework for active hierarchical clustering that repeatedly runs an off-the-shelf clustering algorithm on small subsets of the data and comes with guarantees on performance, measurement complexity and runtime complexity. We instantiate this framework with a simple spectral clustering algorithm and provide concrete results on its performance, showing that, under some assumptions, this algorithm recovers all clusters of size ?(log n) using O(n log^2 n) similarities and runs in O(n log^3 n) time for a dataset of n objects. Through extensive experimentation we also demonstrate that this framework is practically alluring. Clustering is traditionally viewed as an unsupervised method for data analysis. However, in some cases information about the problem domain is available in addition to the data instances themselves. In this paper, we demonstrate how the popular k-means clustering algorithm can be protably modied to make use of this information. In experiments with articial constraints on six data sets, we observe improvements in clustering accuracy. We also apply this method to the real-world problem of automatically detecting road lanes from GPS data and observe dramatic increases in performance. We propose a method of clustering images that combines algorithmic and human input. An algorithm provides us with pairwise image similarities. We then actively obtain selected, more accurate pairwise similarities from humans. A novel method is developed to choose the most useful pairs to show a person, obtaining constraints that improve clustering. In a clustering assignment, elements in each data pair are either in the same cluster or in different clusters. We simulate inverting these pairwise relations and see how that affects the overall clustering. We choose a pair that maximizes the expected change in the clustering. The proposed algorithm has high time complexity, so we also propose a version of this algorithm that is much faster and exactly replicates our original algorithm. We further improve run-time by adding two heuristics, and show that these do not significantly impact the effectiveness of our method. We have run experiments in three different domains, namely leaf, face and scene images, and show that the proposed method improves clustering performance significantly.