On Variants of k-means Clustering

\textit{Clustering problems} often arise in the fields like data mining, machine learning etc. to group a collection of objects into similar groups with respect to a similarity (or dissimilarity) measure. Among the clustering problems, specifically \textit{$k$-means} clustering has got much attention from the researchers. Despite the fact that $k$-means is a very well studied problem its status in the plane is still an open problem. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound in polynomial time is 9+\eps. In this paper, we consider the following variant of $k$-means. Given a set $C$ of points in $\mathcal{R}^d$ and a real $f > 0$, find a finite set $F$ of points in $\mathcal{R}^d$ that minimizes the quantity $f*|F|+\sum_{p\in C} \min_{q \in F} {||p-q||}^2$. For any fixed dimension $d$, we design a local search PTAS for this problem. We also give a "bi-criterion" local search algorithm for $k$-means which uses (1+\eps)k centers and yields a solution whose cost is at most (1+\eps) times the cost of an optimal $k$-means solution. The algorithm runs in polynomial time for any fixed dimension. The contribution of this paper is two fold. On the one hand, we are being able to handle the square of distances in an elegant manner, which yields near optimal approximation bound. This leads us towards a better understanding of the $k$-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that very little is known about the local search method for geometric approximation.Comment: 15 page

Faster K-Means Cluster Estimation

There has been considerable work on improving popular clustering algorithm `K-means' in terms of mean squared error (MSE) and speed, both. However, most of the k-means variants tend to compute distance of each data point to each cluster centroid for every iteration. We propose a fast heuristic to overcome this bottleneck with only marginal increase in MSE. We observe that across all iterations of K-means, a data point changes its membership only among a small subset of clusters. Our heuristic predicts such clusters for each data point by looking at nearby clusters after the first iteration of k-means. We augment well known variants of k-means with our heuristic to demonstrate effectiveness of our heuristic. For various synthetic and real-world datasets, our heuristic achieves speed-up of up-to 3 times when compared to efficient variants of k-means.Comment: 6 pages, Accepted at ECIR 201

Reducing the Time Requirement of k-Means Algorithm

Traditional k-means and most k-means variants are still computationally expensive for large datasets, such as microarray data, which have large datasets with large dimension size d. In k-means clustering, we are given a set of n data points in ddimensional space Rd and an integer k. The problem is to determine a set of k points in Rd, called centers, so as to minimize the mean squared distance from each data point to its nearest center. In this work, we develop a novel k-means algorithm, which is simple but more efficient than the traditional k-means and the recent enhanced k-means. Our new algorithm is based on the recently established relationship between principal component analysis and the k-means clustering. We provided the correctness proof for this algorithm. Results obtained from testing the algorithm on three biological data and six non-biological data (three of these data are real, while the other three are simulated) also indicate that our algorithm is empirically faster than other known k-means algorithms. We assessed the quality of our algorithm clusters against the clusters of a known structure using the Hubert-Arabie Adjusted Rand index (ARIHA). We found that when k is close to d, the quality is good (ARIHA.0.8) and when k is not close to d, the quality of our new k-means algorithm is excellent (ARIHA.0.9). In this paper, emphases are on the reduction of the time requirement of the k-means algorithm and its application to microarray data due to the desire to create a tool for clustering and malaria research. However, the new clustering algorithm can be used for other clustering needs as long as an appropriate measure of distance between the centroids and the members is used. This has been demonstrated in this work on six non-biological data

Fast k-means based on KNN Graph

In the era of big data, k-means clustering has been widely adopted as a basic processing tool in various contexts. However, its computational cost could be prohibitively high as the data size and the cluster number are large. It is well known that the processing bottleneck of k-means lies in the operation of seeking closest centroid in each iteration. In this paper, a novel solution towards the scalability issue of k-means is presented. In the proposal, k-means is supported by an approximate k-nearest neighbors graph. In the k-means iteration, each data sample is only compared to clusters that its nearest neighbors reside. Since the number of nearest neighbors we consider is much less than k, the processing cost in this step becomes minor and irrelevant to k. The processing bottleneck is therefore overcome. The most interesting thing is that k-nearest neighbor graph is constructed by iteratively calling the fast $k$-means itself. Comparing with existing fast k-means variants, the proposed algorithm achieves hundreds to thousands times speed-up while maintaining high clustering quality. As it is tested on 10 million 512-dimensional data, it takes only 5.2 hours to produce 1 million clusters. In contrast, to fulfill the same scale of clustering, it would take 3 years for traditional k-means

Clustering for Different Scales of Measurement - the Gap-Ratio Weighted K-means Algorithm

This paper describes a method for clustering data that are spread out over large regions and which dimensions are on different scales of measurement. Such an algorithm was developed to implement a robotics application consisting in sorting and storing objects in an unsupervised way. The toy dataset used to validate such application consists of Lego bricks of different shapes and colors. The uncontrolled lighting conditions together with the use of RGB color features, respectively involve data with a large spread and different levels of measurement between data dimensions. To overcome the combination of these two characteristics in the data, we have developed a new weighted K-means algorithm, called gap-ratio K-means, which consists in weighting each dimension of the feature space before running the K-means algorithm. The weight associated with a feature is proportional to the ratio of the biggest gap between two consecutive data points, and the average of all the other gaps. This method is compared with two other variants of K-means on the Lego bricks clustering problem as well as two other common classification datasets.Comment: 13 pages, 6 figures, 2 tables. This paper is under the review process for AIAP 201

Document clustering based on firefly algorithm

Document clustering is widely used in Information Retrieval however, existing clustering techniques suffer from local optima problem in determining the k number of clusters.Various efforts have been put to address such drawback and this includes the utilization of swarm-based algorithms such as particle swarm optimization and Ant Colony Optimization.This study explores the adaptation of another swarm algorithm which is the Firefly Algorithm (FA) in text clustering.We present two variants of FA; Weight- based Firefly Algorithm (WFA) and Weight-based Firefly Algorithm II (WFAII).The difference between the two algorithms is that the WFAII, includes a more restricted condition in determining members of a cluster.The proposed FA methods are later evaluated using the 20Newsgroups dataset.Experimental results on the quality of clustering between the two FA variants are presented and are later compared against the one produced by particle swarm optimization, K-means and the hybrid of FA and -K-means. The obtained results demonstrated that the WFAII outperformed the WFA, PSO, K-means and FA-Kmeans. This result indicates that a better clustering can be obtained once the exploitation of a search solution is improved

The k-means algorithm: A comprehensive survey and performance evaluation

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. The k-means clustering algorithm is considered one of the most powerful and popular data mining algorithms in the research community. However, despite its popularity, the algorithm has certain limitations, including problems associated with random initialization of the centroids which leads to unexpected convergence. Additionally, such a clustering algorithm requires the number of clusters to be defined beforehand, which is responsible for different cluster shapes and outlier effects. A fundamental problem of the k-means algorithm is its inability to handle various data types. This paper provides a structured and synoptic overview of research conducted on the k-means algorithm to overcome such shortcomings. Variants of the k-means algorithms including their recent developments are discussed, where their effectiveness is investigated based on the experimental analysis of a variety of datasets. The detailed experimental analysis along with a thorough comparison among different k-means clustering algorithms differentiates our work compared to other existing survey papers. Furthermore, it outlines a clear and thorough understanding of the k-means algorithm along with its different research directions