Gaussian Mean-Shift Is an EM Algorithm

The mean-shift algorithm, based on ideas proposed by Fukunaga and Hosteller, is a hill-climbing algorithm on the density defined by a finite mixture or a kernel density estimate. Mean-shift can be used as a nonparametric clustering method and has attracted recent attention in computer vision applications such as image segmentation or tracking. We show that, when the kernel is Gaussian, mean-shift is an expectation-maximization (EM) algorithm and, when the kernel is non-Gaussian, mean-shift is a generalized EM algorithm. This implies that mean-shift converges from almost any starting point and that, in general, its convergence is of linear order. For Gaussian mean-shift, we show: 1) the rate of linear convergence approaches 0 (superlinear convergence) for very narrow or very wide kernels, but is often close to 1 (thus, extremely slow) for intermediate widths and exactly 1 (sublinear convergence) for widths at which modes merge, 2) the iterates approach the mode along the local principal component of the data points from the inside of the convex hull of the data points, and 3) the convergence domains are nonconvex and can be disconnected and show fractal behavior. We suggest ways of accelerating mean-shift based on the EM interpretation