79 research outputs found
Learning the Irreducible Representations of Commutative Lie Groups
We present a new probabilistic model of compact commutative Lie groups that
produces invariant-equivariant and disentangled representations of data. To
define the notion of disentangling, we borrow a fundamental principle from
physics that is used to derive the elementary particles of a system from its
symmetries. Our model employs a newfound Bayesian conjugacy relation that
enables fully tractable probabilistic inference over compact commutative Lie
groups -- a class that includes the groups that describe the rotation and
cyclic translation of images. We train the model on pairs of transformed image
patches, and show that the learned invariant representation is highly effective
for classification
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
A General Theory of Equivariant CNNs on Homogeneous Spaces
We present a general theory of Group equivariant Convolutional Neural
Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere.
Feature maps in these networks represent fields on a homogeneous base space,
and layers are equivariant maps between spaces of fields. The theory enables a
systematic classification of all existing G-CNNs in terms of their symmetry
group, base space, and field type. We also consider a fundamental question:
what is the most general kind of equivariant linear map between feature spaces
(fields) of given types? Following Mackey, we show that such maps correspond
one-to-one with convolutions using equivariant kernels, and characterize the
space of such kernels
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
Uncertainty-driven Affordance Discovery for Efficient Robotics Manipulation
Robotics affordances, providing information about what actions can be taken
in a given situation, can aid robotics manipulation. However, learning about
affordances requires expensive large annotated datasets of interactions or
demonstrations. In this work, we show active learning can mitigate this problem
and propose the use of uncertainty to drive an interactive affordance discovery
process. We show that our method enables the efficient discovery of visual
affordances for several action primitives, such as grasping, stacking objects,
or opening drawers, strongly improving data efficiency and allowing us to learn
grasping affordances on a real-world setup with an xArm 6 robot arm in a small
number of trials.Comment: Presented at the GMPL workshop @ RSS 202
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