Hessian Sum-Mixtures

A nonlinear least squares Hessian for Gaussian Mixture Factors. Our method explicitly uses the chain rule to take into account the LogSumExp nonlinearity proper to negative log-likelihoods of Gaussian Mixtures. A method to maintain compatibility with standard nonlinear least-squares solvers is provided. This repository contains the companion code and supplementary material for our paper in IEEE Robotics and Automation Letters titled "A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares".

The published article may be found here, and the arXiv version may be found here. The arXiv version also contains the supplementary material.

Getting Started

Install general requirements using

pip install -r requirements.txt

Initialize navlie submodule using

git submodule update --init 

Install the navlie submodule,

pip install -e ./navlie

Install the project library using

pip install -e ./mixtures

Run tests using

cd tests; pytest

This was developed in Python 3.10.1. A virtualenv is recommended.

Reproducing Results From Our Paper

Please run the corresponding bash script,

cd ./scripts/bash_scripts_paper_results
./run_all.sh

Usage

The key functionality of this project has been merged into the navlie library at https://github.com/decargroup/navlie/blob/main/navlie/batch/gaussian_mixtures.py. Errors corresponding to the components are provided to initialize the Gaussian Mixture factors, which then mix the component errors and Jacobians to provide the final mixture error and jacobian. Supplementary material with jacobian derivations is provided in supplementary.pdf.

Citation

If you find this code useful, please consider citing our article,

@ARTICLE{10607873,
  author={Korotkine, Vassili and Cohen, Mitchell and Forbes, James Richard},
  journal={IEEE Robotics and Automation Letters}, 
  title={A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares}, 
  year={2024},
  volume={9},
  number={9},
  pages={7891-7898},
  keywords={Optimization;Simultaneous localization and mapping;Optimization methods;State estimation;Standards;Newton method;Jacobian     matrices;Localization;optimization and optimal control;probabilistic inference;sensor fusion;SLAM},
  doi={10.1109/LRA.2024.3432350}}

License

Distributed under the MIT License. See LICENSE.txt for more information.

Contact

Vassili Korotkine - @decargroup - [email protected]

Project Link: https://github.com/decargroup/hessian_sum_mixtures

Acknowledgments

• The project is built on top of the navlie library for on-manifold state estimation.
• The project is funded via the Natural Sciences and Engineering Research Council of Canada (NSERC) Alliance Grant in collaboration with Denso Corporation.

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