Mode-Finding for Mixtures of Gaussian Distributions

I consider the problem of finding all the modes of a mixture of multivariate Gaussian distributions, which has applications in clustering and regression. I derive exact formulas for the gradient and Hessian and give a partial proof that the number of modes cannot be more than the number of components, and are contained in the convex hull of the component centroids. Then, I develop two exhaustive mode search algorithms: one based on combined quadratic maximisation and gradient ascent and the other one based on a fixed-point iterative scheme. Appropriate values for the search control parameters are derived by taking into account theoretical results regarding the bounds for the gradient and Hessian of the mixture. The significance of the modes is quantified locally (for each mode) by error bars, or confidence intervals (estimated using the values of the Hessian at each mode); and globally by the sparseness of the mixture, measured by its differential entropy (estimated through bounds). I...

• Type: Journal
• Year: 2000
• Where: PAMI
• Authors: Miguel Á. Carreira-Perpiñán
• Keywords: Exhaustive Mode Search, Gradient, Multivariate Gaussian Distributions, PAMI 2000