Advancing Mixture Models for Least Squares Optimization
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.
Comparing Different Approaches
The picture on the left shows the result of optimizing different Gaussian mixture representations from various initial points. While existing Gaussian mixture implementation like Max-Mixture (Olson and Agarwal, 2012) or Sum-Mixture (Rosen et al., 2013) are prone to convergence issues or false minima, the proposed Max-Sum-Mixture is able to converge fast and accurately.
Our paper demonstrates the advantageous performance of Max-Sum-Mixture with several Monte Carlo evaluations and a realistic point set registration. Besides the simulative evaluation, we provide a deep theoretical insight into the connection between the various Gaussian mixture implementations. For further details, we encourage you to read the paper linked below or contact us directly.
Open Source Implementation
We provide implementations of the proposed Max-Sum-Mixture model together with Max-Mixture and Sum-Mixture. Versions for GTSAM as well as for the Ceres solver are available on GitHub:
Publication
- (2021) Advancing Mixture Models for Least Squares Optimization. IEEE Robotics and Automation Letters (RA-L). DOI: 10.1109/LRA.2021.3067307