Harmonic Exponential Families on Manifolds

Taco Cohen, Max Welling
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:1757-1765, 2015.

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 a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applications to Bayesian camera motion estimation (where harmonic exponential families serve as conjugate priors), and modelling of the spatial distribution of earthquakes on the surface of the earth. Our experimental results show that harmonic densities yield a significantly higher likelihood than the best competing method, while being orders of magnitude faster to train.

Cite this Paper

BibTeX
@InProceedings{pmlr-v37-cohenb15, title = {Harmonic Exponential Families on Manifolds}, author = {Cohen, Taco and Welling, Max}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {1757--1765}, year = {2015}, editor = {Bach, Francis and Blei, David}, volume = {37}, series = {Proceedings of Machine Learning Research}, address = {Lille, France}, month = {07--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v37/cohenb15.pdf}, url = {https://proceedings.mlr.press/v37/cohenb15.html}, 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 a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applications to Bayesian camera motion estimation (where harmonic exponential families serve as conjugate priors), and modelling of the spatial distribution of earthquakes on the surface of the earth. Our experimental results show that harmonic densities yield a significantly higher likelihood than the best competing method, while being orders of magnitude faster to train.} }
Endnote
%0 Conference Paper %T Harmonic Exponential Families on Manifolds %A Taco Cohen %A Max Welling %B Proceedings of the 32nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2015 %E Francis Bach %E David Blei %F pmlr-v37-cohenb15 %I PMLR %P 1757--1765 %U https://proceedings.mlr.press/v37/cohenb15.html %V 37 %X 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 a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applications to Bayesian camera motion estimation (where harmonic exponential families serve as conjugate priors), and modelling of the spatial distribution of earthquakes on the surface of the earth. Our experimental results show that harmonic densities yield a significantly higher likelihood than the best competing method, while being orders of magnitude faster to train.
RIS
TY - CPAPER TI - Harmonic Exponential Families on Manifolds AU - Taco Cohen AU - Max Welling BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-cohenb15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 1757 EP - 1765 L1 - http://proceedings.mlr.press/v37/cohenb15.pdf UR - https://proceedings.mlr.press/v37/cohenb15.html AB - 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 a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applications to Bayesian camera motion estimation (where harmonic exponential families serve as conjugate priors), and modelling of the spatial distribution of earthquakes on the surface of the earth. Our experimental results show that harmonic densities yield a significantly higher likelihood than the best competing method, while being orders of magnitude faster to train. ER -
APA
Cohen, T. & Welling, M.. (2015). Harmonic Exponential Families on Manifolds. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:1757-1765 Available from https://proceedings.mlr.press/v37/cohenb15.html.

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