2,844 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
O(4) Expansion of the ladder Bethe-Salpeter equation
The Bethe-Salpeter amplitude is expanded on a hyperspherical basis, thereby
reducing the original 4-dimensional integral equation into an infinite set of
coupled 1-dimensional ones. It is shown that this representation offers a
highly accurate method to determine numerically the bound state solutions. For
generic cases only a few hyperspherical waves are needed to achieve
convergence, both for the ground state as well as for radially or orbitally
excited states. The wave function is reconstructed for several cases and in
particular it is shown that it becomes independent of the relative time in the
nonrelativistic regime.Comment: 21 pages, revte
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
Is the Australian Forex Market Efficient? A Test of the Forward Rate Unbiasness Hypothesis
This paper features a test of the forward rate unbiasedness hypothesis (FRUH), using the Australian dollar with the United States and Japanese currencies using daily frequencies. We evaluate the FRUH on the 1-month forward rate, for both currencies, and the 3-month and 6-month forwards rates for the US dollar only. We adopt a cointegration framework for assessing the FRUH applying a cointegrating VAR model involving Johansen's ML approach. Our results indicate that in all cases the spot and forward rates are integrated of order 1. Furthermore there is evidence of cointegration and in all but one case the cointegrating vector is (1, -1). The error correction term in all cases is statistically significant and has the correct sign.Interest parity, Exchange rates, Market efficiency, Cointegration
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