Advancing Mixture Models for Least Squares Optimization

TL;DR

This paper introduces an exact, nearly linear least squares model for Gaussian mixtures, improving probabilistic estimation in non-Gaussian problems and demonstrating superior performance in registration tasks.

Contribution

It presents a novel, exact least squares representation of Gaussian mixtures that enhances efficiency and accuracy in probabilistic estimation tasks.

Findings

Superior performance in Monte Carlo experiments

Effective in point set registration tasks

Open source implementation available

Abstract

Gaussian mixtures are a powerful and widely used tool to model non-Gaussian estimation problems. They are able to describe measurement errors that follow arbitrary distributions and can represent ambiguity in assignment tasks like point set registration or tracking. However, using them with common least squares solvers is still difficult. Existing approaches are either approximations of the true mixture or prone to convergence issues due to their strong nonlinearity. We propose a novel least squares representation of a Gaussian mixture, which is an exact and almost linear model of the corresponding log-likelihood. Our approach provides an efficient, accurate and flexible model for many probabilistic estimation problems and can be used as cost function for least squares solvers. We demonstrate its superior performance in various Monte Carlo experiments, including different kinds of point set registration. Our implementation is available as open source code for the state-of-the-art solvers Ceres and GTSAM.