Quantum Clustering and Gaussian Mixtures

The mixture of Gaussian distributions, a soft version of k-means ( [2]), is considered a stateof-the-art clustering algorithm. It is widely used in computer vision for selecting classes, e.g., color[4, 1, 5],texture[1, 9], shapes [12, 10]. In this algorithm, each class is described by a Gaussian distribution, defined by its mean and covariance. The data is described by a weighted sum of these Gaussian distributions. We propose a new method, inspired by quantum interference in physics [7]. Instead of modeling each class distribution directly, we model a class wave function such that its magnitude square is the class Gaussian distribution. We then mix the class wave functions to create the mixture wave function. The final mixture distribution is then the magnitude square of the mixture wave function. As a result, we observe the quantum class interference phenomena, not present in the Gaussian mixture model. We show that the quantum method outperforms the Gaussian mixture method in every aspect of the estimations. It provides more accurate estimations of all distribution parameters, with much less fluctuations, and it is also more robust to data deformations from the Gaussian assumptions. We illustrate our method for color segmentation as an example application.