Center-based Clustering Research Articles

A new global optimization method for a symmetric Lipschitz continuous function and the application to searching for a globally optimal partition of a one-dimensional set

In this paper, we consider a global optimization problem for a symmetric Lipschitz continuous function \(g:[a,b]^k\rightarrow {\mathbb {R}}\), whose domain \([a,b]^k\subset {\mathbb {R}}^k\) consists of k! hypertetrahedrons of the same size and shape, in which function g attains equal values. A global minimum can therefore be searched for in one hypertetrahedron only, but then this becomes a global optimization problem with linear constraints. Apart from that, some known global optimization algorithms in standard form cannot be applied to solving the problem. In this paper, it is shown how this global optimization problem with linear constraints can easily be transformed into a global optimization problem on hypercube \([0,1]^k\), for the solving of which an applied DIRECT algorithm in standard form is possible. This approach has a somewhat lower efficiency than known global optimization methods for symmetric Lipschitz continuous functions (such as SymDIRECT or DISIMPL), but, on the other hand, this method allows for the use of publicly available and well developed computer codes for solving a global optimization problem on hypercube \([0,1]^k\) (e.g. the DIRECT algorithm). The method is illustrated and tested on standard symmetric functions and very demanding center-based clustering problems for the data that have only one feature. An application to the image segmentation problem is also shown.

Natural neighbor-based clustering algorithm with local representatives

Clustering by identifying cluster centers is important for detecting patterns in a data set. However, many center-based clustering algorithms cannot process data sets containing non-spherical clusters. In this paper, we propose a novel clustering algorithm called NaNLORE based on natural neighbor and local representatives. Natural neighbor is a new neighbor concept and introduced to compute local density and find local representatives which are points with local maximum density. We first find local representatives and then select cluster centers from the local representatives. The density-adaptive distance is introduced to measure the distance between local representatives, which helps to solve the problem of clustering data sets with complex manifold structure. Cluster centers are characterized by higher density than their neighbors and a relatively large density-adaptive distance from any local representatives with higher density. In experiments, we compare the proposed algorithm NaNLORE with existing algorithms on synthetic and real data sets. Results show that NaNLORE performs better than existing algorithm, especially on clustering non-spherical data and manifold data.

Multiple ellipse fitting by center-based clustering

This paper deals with the multiple ellipse fitting problem based on a given set of data points in a plane. The presumption is that all data points are derived from k ellipses that should be fitted. The problem is solved by means of center-based clustering, where cluster centers are ellipses. If the Mahalanobis distance-like function is introduced in each cluster, then the cluster center is represented by the corresponding Mahalanobis circle-center. The distance from a point $a \in \mathbb{R}^2$ to the Mahalanobis circle is based on the algebraic criterion. The well-known k-means algorithm has been adapted to search for a locally optimal partition of the Mahalanobis circle-centers. Several numerical examples are used to illustrate the proposed algorithm.

Center-based clustering for line detection and application to crop rows detection

This paper proposes a new efficient method for line detection based on known incremental methods of searching for an approximate globally optimal partition of a set of data points A and on the DIRECT algorithm for global optimization. The proposed method was modified for solving the problem of detecting crop rows in agricultural production. This modification can recognize crop rows with a high accuracy, and the corresponding CPU-time is very acceptable. The method has been tested and compared on synthetic data sets with the method based on Hough transformation. The efficiency of this method might be significantly improved in direct application. The proposed method has been used in this paper for the case of two or three crop rows. The generalization to several crop rows is also given in the paper, but was not implemented. Also, the method could be expanded in case when the number of crop rows is not known in advance.

Multiple circle detection based on center-based clustering

The multiple circle detection problem has been considered in the paper on the basis of given data point set A⊂R2. It is supposed that all data points from the set A come from k circles that should be reconstructed or detected. The problem has been solved by the application of center-based clustering of the set A, i.e. an optimal k-partition is searched for, whose clusters are determined by corresponding circle-centers. Thereby, the algebraic distance from a point to the circle is used. First, an adaptation of the well-known k-means algorithm is given in the paper. Also, the incremental algorithm for searching for an approximate globally optimal k-partition is proposed. The algorithm locates either a globally optimal k-partition or a locally optimal k-partition close to the global one. Since optimal partitions with 2, 3, … clusters are determined successively in the algorithm, several well-known indexes for determining an appropriate number of clusters in a partition are adopted for this case. Thereby, the Hausdorff distance between two circles is used and adopted. The proposed method and algorithm are illustrated and tested on several numerical examples.

ProDomAs, protein domain assignment algorithm using center-based clustering and independent dominating set

Decomposition of structural domains is an essential task in classifying protein structures, predicting protein function, and many other proteomics problems. As the number of known protein structures in PDB grows exponentially, the need for accurate automatic domain decomposition methods becomes more essential. In this article, we introduce a bottom-up algorithm for assigning protein domains using a graph theoretical approach. This algorithm is based on a center-based clustering approach. For constructing initial clusters, members of an independent dominating set for the graph representation of a protein are considered as the centers. A distance matrix is then defined for these clusters. To obtain final domains, these clusters are merged using the compactness principle of domains and a method similar to the neighbor-joining algorithm considering some thresholds. The thresholds are computed using a training set consisting of 50 protein chains. The algorithm is implemented using C++ language and is named ProDomAs. To assess the performance of ProDomAs, its results are compared with seven automatic methods, against five publicly available benchmarks. The results show that ProDomAs outperforms other methods applied on the mentioned benchmarks. The performance of ProDomAs is also evaluated against 6342 chains obtained from ASTRAL SCOP 1.71. ProDomAs is freely available at http://www.bioinf.cs.ipm.ir/software/prodomas.

Data clustering for circle detection

This paper considers multiple-circle detection problem on the basis of given data. The problem is being solved by application of center-based clustering method. For the purpose of searching a locally optimal partition, modeled on a well-known k-means algorithm, k-closest circles algorithm has been constructed. The method has been illustrated with several numerical examples. Key words: data clustering, circle detection, k -means, locally optimal partition

Center-based clustering under perturbation stability

Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial (2010) [11] suggested an approach aimed at bypassing this computational barrier by using properties of instances one might hope to hold in practice. In particular, they argue that instances in practice should be stable to small perturbations in the metric space and give an efficient algorithm for clustering instances of the Max-Cut problem that are stable to perturbations of size O ( n 1 / 2 ) . In addition, they conjecture that instances stable to as little as O ( 1 ) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any center-based clustering objective (such as k-median, k-means, and k-center). Specifically, we show we can efficiently find the optimal clustering assuming only stability to factor-3 perturbations of the underlying metric in spaces without Steiner points, and stability to factor 2 + 3 perturbations for general metrics. In particular, we show for such instances that the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. We also present NP-hardness results under a weaker but related condition.

Application of shuffled frog-leaping algorithm on clustering

Evolutionary algorithms, such as shuffled frog leaping, are stochastic search methods that mimic natural biological evolution and/or the social behavior of species. Such algorithms have been developed to arrive at near-optimum solutions to complex and large-scale optimization problems which cannot be solved by gradient-based mathematical programming techniques. The shuffled frog-leaping algorithm draws its formulation from two other search techniques: the local search of the “particle swarm optimization” technique and the competitiveness mixing of information of the “shuffled complex evolution” technique. Cluster analysis is one of the attractive data mining techniques which is used in many fields. One popular class of data clustering algorithms is the center-based clustering algorithm. K-means is used as a popular clustering method due to its simplicity and high speed in clustering large datasets. However, k-means has two shortcomings: Dependency on the initial state and convergence to local optima and global solutions of large problems cannot be found with reasonable amount of computation effort. In order to overcome local optima problem, lots of studies are done in clustering. In this paper, we proposed an application of shuffled frog-leaping algorithm in clustering (SFLK-means). We compared SFLK-means with other heuristics algorithm in clustering, such as GAK, SA, TS, and ACO, by implementing them on several simulations and real datasets. Our finding shows that the proposed algorithm works better than others.

A Unified Continuous Optimization Framework for Center-Based Clustering Methods

Center-based partitioning clustering algorithms rely on minimizing an appropriately formulated objective function, and different formulations suggest different possible algorithms. In this paper, w...