A continuation approach to mode-finding of multivariate Gaussian mixtures and kernel density estimates

Abstract

Gaussian mixtures (i.e. linear combinations of multivariate Gaussian probability densities) appear in numerous applications due to their universal ability to approximate multimodal probability distributions. Finding the modes (maxima) of a Gaussian mixture is a fundamental problem arising in many practical applications such as machine learning and digital image processing. In this paper, we propose a computationally efficient method for finding a significant mode of the Gaussian mixture. Such a mode represents an area of large probability, and it often coincides with the global mode of the mixture. The proposed method uses a Gaussian convolution in order to remove undesired local maxima of the Gaussian mixture and preserve its underlying structure. The transformation between the maximizers of the smoothed Gaussian mixture and the original one is formulated as a differential equation. A robust trust region method for tracing the solution curve of this differential equation is described. Our formulation also allows mixtures with negative weights or even negative values, which occur in some applications related to machine learning or quantum mechanics. The applicability of the method to mode-finding of Gaussian kernel density estimates obtained from experimental data is illustrated. Finally, some numerical results are given to demonstrate the ability of the method to find significant modes of Gaussian mixtures and kernel density estimates.