Distributed Approximation of Joint Measurement Distributions Using Mixtures of Gaussians

This paper presents an approach to distributively approximate the continuous probability distribution that describes the fusion of sensor measurements from many networked robots. Each robot forms a weighted mixture of Gaussians to represent the measurement distribution of its local observation. From this mixture set, the robot then draws samples of Gaussian elements to enable the use of a consensus-based algorithm that evolves the corresponding canonical parameters. We show that the these evolved parameters describe a distribution that converges weakly to the joint of all the robots' unweighted mixture distributions, which itself converges weakly to the joint measurement distribution as more system resources are allocated. The major innovation of this approach is to combine sample-based sensor fusion with the notion of pre-convergence termination that results in scalable multi-robot system. We also derive bounds and convergence rates for the approximated joint measurement distribution, specifically the elements of its information vectors and the eigenvalues of its information matrices. Most importantly, these performance guarantees do not come at a cost of complexity, since computational and communication complexity scales quadratically with respect to the Gaussian dimension, linearly with respect to the number of samples, and constant with respect to the number of robots. Results from numerical simulations for object localization are discussed using both Gaussians and mixtures of Gaussians.