36 research outputs found
Harmonic Exponential Families on Manifolds
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.Comment: fixed typ
Transformation Properties of Learned Visual Representations
When a three-dimensional object moves relative to an observer, a change
occurs on the observer's image plane and in the visual representation computed
by a learned model. Starting with the idea that a good visual representation is
one that transforms linearly under scene motions, we show, using the theory of
group representations, that any such representation is equivalent to a
combination of the elementary irreducible representations. We derive a striking
relationship between irreducibility and the statistical dependency structure of
the representation, by showing that under restricted conditions, irreducible
representations are decorrelated. Under partial observability, as induced by
the perspective projection of a scene onto the image plane, the motion group
does not have a linear action on the space of images, so that it becomes
necessary to perform inference over a latent representation that does transform
linearly. This idea is demonstrated in a model of rotating NORB objects that
employs a latent representation of the non-commutative 3D rotation group SO(3).Comment: T.S. Cohen & M. Welling, Transformation Properties of Learned Visual
Representations. In International Conference on Learning Representations
(ICLR), 201
Gauge Equivariant Convolutional Networks and the Icosahedral CNN
The principle of equivariance to symmetry transformations enables a
theoretically grounded approach to neural network architecture design.
Equivariant networks have shown excellent performance and data efficiency on
vision and medical imaging problems that exhibit symmetries. Here we show how
this principle can be extended beyond global symmetries to local gauge
transformations. This enables the development of a very general class of
convolutional neural networks on manifolds that depend only on the intrinsic
geometry, and which includes many popular methods from equivariant and
geometric deep learning. We implement gauge equivariant CNNs for signals
defined on the surface of the icosahedron, which provides a reasonable
approximation of the sphere. By choosing to work with this very regular
manifold, we are able to implement the gauge equivariant convolution using a
single conv2d call, making it a highly scalable and practical alternative to
Spherical CNNs. Using this method, we demonstrate substantial improvements over
previous methods on the task of segmenting omnidirectional images and global
climate patterns.Comment: Proceedings of the International Conference on Machine Learning
(ICML), 201