K-means clustering is a very popular clustering technique, which is used in numerous applications. Given a set of n data points in R d and an integer k, the problem is to determine a set of k points R d , called centers, so as to minimize the mean squared distance from each data point to its nearest center. A popular heuristic for k-means clustering is Lloyd's algorithm. In this paper we present a simple and efficient implementation of Lloyd's k-means clustering algorithm, which we call the filtering algorithm. This algorithm is very easy to implement. It differs from most other approaches in that it precomputes a kd-tree data structure for the data points rather than the center points. We establish the practical efficiency of the filtering algorithm in two ways. First, we present a data-sensitive analysis of the algorithm's running time. Second, we have implemented the algorithm and performed a number of empirical studies, both on synthetically generated data and on real...

Clustering is a common unsupervised learning technique used to discover group structure in a set of data. While there exist many algorithms for clustering, the important issue of feature selection, that is, what attributes of the data should be used by the clustering algorithms, is rarely touched upon. Feature selection for clustering is difficult because, unlike in supervised learning, there are no class labels for the data and, thus, no obvious criteria to guide the search. Another important problem in clustering is the determination of the number of clusters, which clearly impacts and is influenced by the feature selection issue. In this paper, we propose the concept of feature saliency and introduce an expectation-maximization (EM) algorithm to estimate it, in the context of mixture-based clustering. Due to the introduction of a minimum message length model selection criterion, the saliency of irrelevant features is driven toward zero, which corresponds to performing feature selection. The criterion and algorithm are then extended to simultaneously estimate the feature saliencies and the number of clusters.

Abstract—DNA microarray technology has now made it possible to simultaneously monitor the expression levels of thousands of genes during important biological processes and across collections of related samples. Elucidating the patterns hidden in gene expression data offers a tremendous opportunity for an enhanced understanding of functional genomics. However, the large number of genes and the complexity of biological networks greatly increases the challenges of comprehending and interpreting the resulting mass of data, which often consists of millions of measurements. A first step toward addressing this challenge is the use of clustering techniques, which is essential in the data mining process to reveal natural structures and identify interesting patterns in the underlying data. Cluster analysis seeks to partition a given data set into groups based on specified features so that the data points within a group are more similar to each other than the points in different groups. A very rich literature on cluster analysis has developed over the past three decades. Many conventional clustering algorithms have been adapted or directly applied to gene expression data, and also new algorithms have recently been proposed specifically aiming at gene expression data. These clustering algorithms have been proven useful for identifying biologically relevant groups of genes and samples. In this paper, we first briefly introduce the concepts of microarray technology and discuss the basic elements of clustering on gene expression data. In particular, we divide cluster analysis for gene expression data into three categories. Then, we present specific challenges pertinent to each clustering category and introduce several representative approaches. We also discuss the problem of cluster validation in three aspects and review various methods to assess the quality and reliability of clustering results. Finally, we conclude this paper and suggest the promising trends in this field. Index Terms—Microarray technology, gene expression data, clustering.

Abstract—In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data, subject to a given distortion. By analyzing the coding length/rate of mixed data, we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate-distortion theory. We show that a deterministic segmentation is approximately the (asymptotically) optimal solution for compressing mixed data. We propose a very simple and effective algorithm that depends on a single parameter, the allowable distortion. At any given distortion, the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation. Simulation results reveal intriguing phase-transition-like behaviors of the number of segments when changing the level of distortion or the amount of outliers. Finally, we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data. Index Terms—Multivariate mixed data, data segmentation, data clustering, rate distortion, lossy coding, lossy compression, image segmentation, microarray data clustering. 1

Document clustering techniques mostly rely on single term analysis of the document data set, such as the Vector Space Model. To achieve more accurate document clustering, more informative features including phrases and their weights are particularly important in such scenarios. Document clustering is particularly useful in many applications such as automatic categorization of documents, grouping search engine results, building a taxonomy of documents, and others. This paper presents two key parts of successful document clustering. The first part is a novel phrase-based document index model, the Document Index Graph, which allows for incremental construction of a phrase-based index of the document set with an emphasis on efficiency, rather than relying on single-term indexes only. It provides efficient phrase matching that is used to judge the similarity between documents. The model is flexible in that it could revert to a compact representation of the vector space model if we choose not to index phrases. The second part is an incremental document clustering algorithm based on maximizing the tightness of clusters by carefully watching the pair-wise document similarity distribution inside clusters. The combination of these two components creates an underlying model for robust and accurate document similarity calculation that leads to much improved results in Web document clustering over traditional methods.

Abstract—For several major applications of data analysis, objects are often not represented as feature vectors in a vector space, but rather by a matrix gathering pairwise proximities. Such pairwise data often violates metricity and, therefore, cannot be naturally embedded in a vector space. Concerning the problem of unsupervised structure detection or clustering, in this paper, a new embedding method for pairwise data into Euclidean vector spaces is introduced. We show that all clustering methods, which are invariant under additive shifts of the pairwise proximities, can be reformulated as grouping problems in Euclidian spaces. The most prominent property of this constant shift embedding framework is the complete preservation of the cluster structure in the embedding space. Restating pairwise clustering problems in vector spaces has several important consequences, such as the statistical description of the clusters by way of cluster prototypes, the generic extension of the grouping procedure to a discriminative prediction rule, and the applicability of standard preprocessing methods like denoising or dimensionality reduction. Index Terms—Clustering, pairwise proximity data, cost function, embedding, MDS. 1

Abstract — Computer vision is a broad and significant ongoing research challenge, even when performed on an individual image or on streaming video from a high-quality stationary camera with abundant computational resources. When faced with streaming video from a lower-quality, rapidly moving camera and limited computational resources, the challenge increases. We present our implementation of a vision system on a mobile robot platform that uses a camera image as the primary sensory input. Having to perform all processing, including segmentation and object detection, in real-time on-board the robot, eliminates the possibility of using some state-of-the-art methods that otherwise might apply. We describe the methods that we developed to achieve a practical vision system within these constraints. Our approach is fully implemented and tested on a team of Sony AIBO robots. Index Terms — Vision and Recognition, Legged Robots. I.

Clustering ensembles have emerged as a powerful method for improving both the robustness as well as the stability of unsupervised classification solutions. However, finding a consensus clustering from multiple partitions is a difficult problem that can be approached from graph-based, combinatorial or statistical perspectives. This study extends previous research on clustering ensembles in several respects. First, we introduce a unified representation for multiple clusterings and formulate the corresponding categorical clustering problem. Second, we propose a probabilistic model of consensus using a finite mixture of multinomial distributions in a space of clusterings. A combined partition is found as a solution to the corresponding maximum likelihood problem using the EM algorithm. Third, we define a new consensus function that is related to the classical intra-class variance criterion using the generalized mutual information definition. Finally, we demonstrate the efficacy of combining partitions generated by weak clustering algorithms that use data projections and random data splits. A simple explanatory model is offered for the behavior of combinations of such weak clustering components. Combination accuracy is analyzed as a function of several parameters that control the power and resolution of component partitions as well as the number of partitions. We also analyze clustering ensembles with incomplete information and the effect of missing cluster labels on the quality of overall consensus. Experimental results demonstrate the effectiveness of the proposed methods on several real-world datasets.

Abstract—Given a data set P,ak-means query returns k points in space (called centers), such that the average squared distance between each point in P and its nearest center is minimized. Since this problem is NP-hard, several approximate algorithms have been proposed and used in practice. In this paper, we study continuous k-means computation at a server that monitors a set of moving objects. Reevaluating k-means every time there is an object update imposes a heavy burden on the server (for computing the centers from scratch) and the clients (for continuously sending location updates). We overcome these problems with a novel approach that significantly reduces the computation and communication costs, while guaranteeing that the quality of the solution, with respect to the reevaluation approach, is bounded by a user-defined tolerance. The proposed method assigns each moving object a threshold (i.e., range) such that the object sends a location update only when it crosses the range boundary. First, we develop an efficient technique for maintaining the k-means. Then, we present mathematical formulas and algorithms for deriving the individual thresholds. Finally, we justify our performance claims with extensive experiments. Index Terms—k-Means, continuous monitoring, query processing. Ç

Abstract—Nearest neighbor search and many other numerical data analysis tools most often rely on the use of the euclidean distance. When data are high dimensional, however, the euclidean distances seem to concentrate; all distances between pairs of data elements seem to be very similar. Therefore, the relevance of the euclidean distance has been questioned in the past, and fractional norms (Minkowski-like norms with an exponent less than one) were introduced to fight the concentration phenomenon. This paper justifies the use of alternative distances to fight concentration by showing that the concentration is indeed an intrinsic property of the distances and not an artifact from a finite sample. Furthermore, an estimation of the concentration as a function of the exponent of the distance and of the distribution of the data is given. It leads to the conclusion that, contrary to what is generally admitted, fractional norms are not always less concentrated than the euclidean norm; a counterexample is given to prove this claim. Theoretical arguments are presented, which show that the concentration phenomenon can appear for real data that do not match the hypotheses of the theorems, in particular, the assumption of independent and identically distributed variables. Finally, some insights about how to choose an optimal metric are given. Index Terms—Nearest neighbor search, high-dimensional data, distance concentration, fractional distances. 1