Julien Ugon

Fast modified global k-means algorithm for incremental cluster construction

The k-means algorithm and its variations are known to be fast clustering algorithms. However, they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets. Recently, incremental approaches have been developed to resolve difficulties with the choice of starting points. The global k-means and the modified global k-means algorithms are based on such an approach. They iteratively add one cluster center at a time. Numerical experiments show that these algorithms considerably improve the k-means algorithm. However, they require storing the whole affinity matrix or computing this matrix at each iteration. This makes both algorithms time consuming and memory demanding for clustering even moderately large datasets. In this paper, a new version of the modified global k-means algorithm is proposed. We introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset. We exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage. Results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k-means algorithms.

Coverage in WLAN with Minimum Number of Access Points

When designing wireless communication systems, it is very important to know the optimum numbers and locations for the access points (APs). The impact of incorrect placement of APs is significant. Placing too many access points increases the cost of deployment and interference, while placing too few access points can lead to coverage gaps. In this paper we describe a novel mathematical model developed to find the optimal number of APs and their locations in an environment that includes obstacles. To solve the problem, we use a new global optimization (AGOP) algorithm. The results obtained indicate that our model and software is able to solve optimal coverage problems for a design area with different number of users

Optimal placement of access point in WLAN based on a new algorithm

When designing wireless communication systems, it is very important to know the optimum numbers and locations for the access points (APs). The impact of incorrect placement of APs is significant. If they are placed too far apart, they will generate a coverage gap, but if they are too close to each other, this will lead to excessive co-channel interferences. In this paper we describe a mathematical model developed to find the optimal number and location of APs. To solve the problem, we use the discrete gradient optimization algorithm developed at the University of Ballarat. Results indicate that our model is able to solve optimal coverage problems for different numbers of users.

Implementation of novel methods of global and nonsmooth optimization: GANSO programming library

We discuss the implementation of a number of modern methods of global and nonsmooth continuous optimization, based on the ideas of Rubinov, in a programming library GANSO. GANSO implements the derivative-free bundle method, the extended cutting angle method, dynamical system-based optimization and their various combinations and heuristics. We outline the main ideas behind each method, and report on the interfacing with Matlab and Maple packages. §We dedicate this contribution to the memory of our dear friend and teacher Alex Rubinov, who was an inspiration for so many people around the globe.

Piecewise Partially Separable Functions and a Derivative-free Algorithm for Large Scale Nonsmooth Optimization

This paper introduces the notion of piecewise partially separable functions and studies their properties. We also consider some of many applications of these functions. Finally, we consider the problem of minimizing of piecewise partially separable functions and develop an algorithm for its solution. This algorithm exploits the structure of such functions. We present the results of preliminary numerical experiments.

An algorithm for minimizing clustering functions

The problem of cluster analysis is formulated as a problem of nonsmooth, nonconvex optimization. An algorithm for solving the latter optimization problem is developed which allows one to significantly reduce the computational efforts. This algorithm is based on the so-called discrete gradient method. Results of numerical experiments are presented which demonstrate the effectiveness of the proposed algorithm.

Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems

Abstract This paper introduces an algorithm for solving the minimum sum-of-squares clustering problems using their difference of convex representations. A non-smooth non-convex optimization formulation of the clustering problem is used to design the algorithm. Characterizations of critical points, stationary points in the sense of generalized gradients and inf-stationary points of the clustering problem are given. The proposed algorithm is tested and compared with other clustering algorithms using large real world data sets.

Nonsmooth nonconvex optimization approach to clusterwise linear regression problems

Clusterwise regression consists of finding a number of regression functions each approximating a subset of the data. In this paper, a new approach for solving the clusterwise linear regression problems is proposed based on a nonsmooth nonconvex formulation. We present an algorithm for minimizing this nonsmooth nonconvex function. This algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate a good starting point for solving global optimization problems at each iteration of the incremental algorithm. Such an approach allows one to find global or near global solution to the problem when the data sets are sufficiently dense. The algorithm is compared with the multistart Spath algorithm on several publicly available data sets for regression analysis.

Classification through incremental max–min separability

Piecewise linear functions can be used to approximate non-linear decision boundaries between pattern classes. Piecewise linear boundaries are known to provide efficient real-time classifiers. However, they require a long training time. Finding piecewise linear boundaries between sets is a difficult optimization problem. Most approaches use heuristics to avoid solving this problem, which may lead to suboptimal piecewise linear boundaries. In this paper, we propose an algorithm for globally training hyperplanes using an incremental approach. Such an approach allows one to find a near global minimizer of the classification error function and to compute as few hyperplanes as needed for separating sets. We apply this algorithm for solving supervised data classification problems and report the results of numerical experiments on real-world data sets. These results demonstrate that the new algorithm requires a reasonable training time and its test set accuracy is consistently good on most data sets compared with mainstream classifiers.

Classes and clusters in data analysis

We discuss the relation between classes and clusters in datasets with given classes. We examine the distribution of classes within obtained clusters, using different clustering methods which are based on different techniques. We also study the structure of the obtained clusters. One of the main conclusions, obtained in this research is that the notion purity cannot be always used for evaluation of accuracy of clustering techniques.