Clustering Optimization: Finding the Number and Centroids of Clusters by a Fourier-based Algorithm

We propose a Fourier-based approach for optimization of several clustering algorithms. Mathematically, clusters data can be described by a density function represented by the Dirac mixture distribution. The density function can be smoothed by applying the Fourier transform and a Gaussian filter. The determination of the optimal standard deviation of the Gaussian filter will be accomplished by the use of a convergence criterion related to the correlation between the smoothed and the original density functions. In principle, the optimal smoothed density function exhibits local maxima, which correspond to the cluster centroids. Thus, the complex task of finding the centroids of the clusters is simplified by the detection of the peaks of the smoothed density function. A multiple sliding windows procedure is used to detect the peaks. The remarkable accuracy of the proposed algorithm demonstrates its capability as a reliable general method for enhancement of the clustering performance, its global optimization and also removing the initialization problem in many clustering methods.