Contributions to k-means clustering and regression via classification algorithms

The dissertation deals with clustering algorithms and transforming regression prob-lems into classification problems. The main contributions of the dissertation are twofold; first, to improve (speed up) the clustering algorithms and second, to develop a strict learn-ing environment for solving regression problems as classification tasks by using support vector machines (SVMs). An extension to the most popular unsupervised clustering method, k-means algorithm, is proposed, dubbed k-means 2 (k-means squared) algorithm, applicable to ultra large datasets. The main idea is based on using a small portion of the dataset in the first stage of the clustering. Thus, the centers of such a smaller dataset are computed much faster than if computing the centers based on the whole dataset. These final centers of the first stage are naturally much closer to the locations of the final centers rendering a great reduction in the total computational cost. For large datasets the speed up in computation exhibited a trend which is shown to be high and rising with the increase in the size of the dataset. The total transient time for the fast stage was found to depend largely on the portion of the dataset selected in the stage. For medium size datasets it has been shown that an 8-10% portion of data used in the fast stage is a reasonable choice. The centers of the 8-10% samples computed during the fast stage may oscillate towards the final centers' positions of the fast stage along the centers' movement path. The slow stage will start with the final centers of the fast phase and the paths of the centers in the second stage will be much shorter than the ones of a classic k-means algorithm. Additionally, the oscillations of the slow stage centers' trajectories along the path to the final centers' positions are also greatly minimized. In the second part of the dissertation, a novel approach of posing a solution of regression problems as the multiclass classification tasks within the common framework of kernel machines is proposed. Based on such an approach both the nonlinear (NL) regression problems and NL multiclass classification tasks will be solved as multiclass classification problems by using SVMs. The accuracy of an approximating classification (hyper)Surface (averaged over several benchmarking data sets used in this study) to the data points over a given high-dimensional input space created by a nonlinear multiclass classifier is slightly superior to the solution obtained by regression (hyper)Surface. In terms of the CPU time needed for training (i.e. for tuning the hyperparameters of the models), the nonlinear SVM classifier also shows significant advantages. Here, the comparisons between the solutions obtained by an SVM solving given regression problem as a classic SVM regressor and as the SVM classifier have been performed. In order to transform a regression problem into a classification task, four possible discretizations of a continuous output (target) vector y are introduced and compared. A very strict double (nested) cross-validation technique has been used for measuring the performances of regression and multiclass classification SVMs. In order to carry out fair comparisons, SVMs are used for solving both tasks - regression and multiclass classification. The readily available and most popular benchmarking SVM tool, LibSVM, was used in all experiments. The results in solving twelve benchmarking regression tasks shown here will present SVM regression and classification algorithms as strongly competing models where each approach shows merits for a specific class of high-dimensional function approximation problems.