Riemannian Gaussian Processes

Disclaimer

This is supplementary code for the paper "Matern Gaussian processes on Riemannian manifolds". Please note that the code is provided only as a demo. The demo shows the example of a Gaussian process regression on the dragon manifold with a Matern kernel defined on the manifold. Additional info can be found in the paper and in the Appendix A.

Preliminaries

We use and have run our code on Python 3.

The code relies heavily on the Firedrake package. In order to run our demo, please follow the installation instructions of the Firedrake. Note that Firedrake may take some time to install.

You should also have paramz, autograd and networkx installed inside firedrake virtual environment. Simply run

pip install -e .

with firedrake virtual environment activated. This will install the manifold_matern package and the dependencies.

Usage

To run our demo, activate firedrake virtual environment first by typing

source firedrake/bin/activate

in your terminal.

Then simply enter

python demo_dragon.py

This will create a directory output where a number of .pvd-files will be saved such as files with the ground truth function, mean and standard deviation of the posterior GP, and a bunch of files with posterior samples (16 by default). The .pvd-files can be viewed, for example, with Paraview.

Note that depending on your computing power, several computationally heavy parts of the code may take a while to run.

If you wish, you can also obtain images like ones provided in the paper that you can see with your favorite image viewer by providing an option

python demo_dragon.py --mayavi

Note that that requires Mayavi to be properly installed. If mayavi is not installed, providing this option will not have any effect.

You can also play with the script by providing several options:

python demo_dragon.py --help
usage: demo_dragon.py [-h] [--num-eigenpairs NUM_EIGENPAIRS] [--seed SEED]
                      [--output-dir OUTPUT_DIR]
                      [--eigenpairs-file EIGENPAIRS_FILE] [--mayavi]
                      [--num-samples NUM_SAMPLES]

optional arguments:
  -h, --help            show this help message and exit
  --num-eigenpairs NUM_EIGENPAIRS
                        Number of eigenpairs to use. Default is 500
  --seed SEED           Random seed
  --output-dir OUTPUT_DIR
                        Output directory to save .pvd files to. Default is
                        ./output
  --eigenpairs-file EIGENPAIRS_FILE
                        .npy file with precomputed eigenpairs
  --mayavi              Render results to .png with Mayavi
  --num-samples NUM_SAMPLES
                        Number of random samples to generate

Issues with mayavi

If you run into issues with mayavi rendering using PyQt backend you may want to use wxPython backend instead. See a corresponing issue on mayavi github. This issue may arise with the Ubuntu 18.04 Linux distribution. Installing wxPython in firedrake virtual environment and setting

export ETS_TOOLKIT=wx

should fix this.

Library usage

You can also use manifold_matern as a library to train a GP on a mesh. The library provides ManifoldMaternGP class:

>>> from  manifold_matern import ManifoldMaternGP
>>> gp = ManifoldMaternGP(mesh, V, X, Y, eigenpairs)
>>> gp.optimize()

Please refer to the documentation for details and to our demo as an example.

Citing

@article{borovitskiy2020,
    title={Matern Gaussian processes on Riemannian manifolds},
    author={Viacheslav Borovitskiy and Alexander Terenin and Peter Mostowsky and Marc Peter Deisenroth},
    booktitle={Advances in Neural Information Processing Systems},
    year={2020}}