Abstract

In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponential family distributions on manifolds such as the torus, sphere, and rotation groups, and show that for these distributions the gradient of the log-likelihood can be computed efficiently using generalized Fast Fourier Transforms. We discuss applications to Bayesian transformation estimation (where harmonic exponential families appear as conjugate priors to a special parameterization of the normal distribution), and modelling of the spatial distribution of earthquakes on the surface of the earth.

The second part of this talk is about ongoing work on Group-equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convnets that can deal with geometrical variability due to Lie groups. By convolving over groups G larger than the translation group, G-CNNs build representations that are equivariant to these groups, which allows for a much greater degree of parameter sharing. Implemented naively, the group convolutions employed by G-CNNs are very slow, so we introduce a fast method based on the Fast Fourier Transform on G that computes both the forward and backward propagation through a G-CNN efficiently.