ReLie

Official repository to reproduce experiments for the AISTATS-19 publication on Reparameterizing Distributions over Lie Groups [1] (arxiv). For a more intuitive understanding of our work take a look at the presentation slides prepared for our talk in Okinawa.

From left to right, examples of SO(2), SO(3), and SE(3) group actions.

Implementation

We implement the code for SO(3) in PyTorch by building on the torch.distributions.transform framework. We extend this framework, as the Lie group exponential map is not a bijection, but a locally invertible function / a local diffeomorphism:

• relie.LocalDiffeoTransform generalizes torch.distributions.transform.Transform
• relie.LocalDiffeoTransformedDistribution generalizes torch.distributions.transform.TransformedDistribution

The simplest way of creating a distribution on the group, is by putting a zero-mean Gaussian on the algebra, pushing this forward and left-multiplying with a group element, to put the 'mean' of the resulting distribution away from the identity. This can be constructed as follows:

from relie import (
    SO3ExpTransform,
    SO3MultiplyTransform,
    LocalDiffeoTransformedDistribution as LDTD,
)

alg_loc = ...  # of shape [batch, 3], dtype=double
scale = ...  # of shape [batch, 3], dtype=double
loc = so3_exp(alg_loc)  # of shape [batch, 3, 3]

alg_distr = Normal(torch.zeros_like(scale), scale)
transforms = [SO3ExpTransform(k_max=3), SO3MultiplyTransform(loc)]
group_distr = LDTD(alg_distr, transforms)

This can then be used for e.g. Variational Inference:

z = group_distr.rsample()
entropy = -group_distr.log_prob(z)

Note:

• We require double precision.
• We consider 2 * k_max + 1 pre-images. In our experience, k_max=3 is sufficient.
• Parametrizing the mean with an algebra element that is mapped to the group with the exponential map should not be used in the context of auto-encoders. See [2, 3] for details.

LI-Flow

Alternatively, one can construct a NICE-style normalizing flow. See relie.experiments.so3_multimodal_flow for an example.

Experiments

Please find the experiments of the paper in the package relie.experiments.

Contact

For comments and questions regarding this repository, feel free to reach out to Pim de Haan.

License

MIT

References

[1] Falorsi, L., de Haan, P., Davidson, T. & Forré, P.
Reparametrizing Distributions on Lie Groups
AISTATS (2019)
[2] Falorsi, L., de Haan, P., Davidson, T. R., De Cao, N., Weiler, M., Forré, P., & Cohen, T. S.
Explorations in homeomorphic variational auto-encoding
ICML 2018 workshop on Theoretical Foundations and Applications of Deep Generative Models (2018)
[3] de Haan, P., and Falorsi, L..
Topological Constraints on Homeomorphic Auto-Encoding
NeurIPS 2018 workshop on Integration of Deep Learning Theories (2018)

BibTeX format:

@article{falorsi2019reparameterizing,
  title={Reparameterizing distributions on Lie groups},
  author={Falorsi, L. and
          de Haan, P. and
          Davidson, T.R. and
          Forr{\'e}, P.},
  journal={22nd International Conference on Artificial Intelligence and Statistics (AISTATS-19)},
  year={2019}
}