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 submission to IEEE Robotics and Automation Letters titled "A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares".

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 is in the gaussian_mixtures.py file in the mixtures library. 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 arxiv submission,

@misc{korotkine2024hessian,
      title={A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares}, 
      author={Vassili Korotkine and Mitchell Cohen and James Richard Forbes},
      year={2024},
      eprint={2404.05452},
      archivePrefix={arXiv},
      primaryClass={cs.RO}
}

License

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

Contact

Vassili Korotkine - @decargroup - vassili.korotkine@mail.mcgill.ca

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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