1,279 research outputs found
Integral Geometric Dual Distributions of Multilinear Models
We propose an integral geometric approach for computing dual distributions
for the parameter distributions of multilinear models. The dual distributions
can be computed from, for example, the parameter distributions of conics,
multiple view tensors, homographies, or as simple entities as points, lines,
and planes. The dual distributions have analytical forms that follow from the
asymptotic normality property of the maximum likelihood estimator and an
application of integral transforms, fundamentally the generalised Radon
transforms, on the probability density of the parameters. The approach allows
us, for instance, to look at the uncertainty distributions in feature
distributions, which are essentially tied to the distribution of training data,
and helps us to derive conditional distributions for interesting variables and
characterise confidence intervals of the estimates
Measuring the interactions among variables of functions over the unit hypercube
By considering a least squares approximation of a given square integrable
function by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of . This definition extends the concept of Banzhaf interaction
index introduced in cooperative game theory. Our approach is partly inspired
from multilinear regression analysis, where interactions among the independent
variables are taken into consideration. We show that this interaction index has
appealing properties which naturally generalize the properties of the Banzhaf
interaction index. In particular, we interpret this index as an expected value
of the difference quotients of or, under certain natural conditions on ,
as an expected value of the derivatives of . These interpretations show a
strong analogy between the introduced interaction index and the overall
importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a
few applications of the interaction index
Hochschild Cohomology Theories in White Noise Analysis
We show that the continuous Hochschild cohomology and the differential
Hochschild cohomology of the Hida test algebra endowed with the normalized Wick
product are the same.Comment: This is a contribution to the Special Issue on Deformation
Quantization, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?
We discuss of the conceptual difficulties connected with the
anticommutativity of classical fermion fields, and we argue that the "space" of
all classical configurations of a model with such fields should be described as
an infinite-dimensional supermanifold M.
We discuss the two main approaches to supermanifolds, and we examine the
reasons why many physicists tend to prefer the Rogers approach although the
Berezin-Kostant-Leites approach is the more fundamental one. We develop the
infinite-dimensional variant of the latter, and we show that the functionals on
classical configurations considered in a previous paper are nothing but
superfunctions on M. We present a programme for future mathematical work, which
applies to any classical field model with fermion fields. This programme is
(partially) implemented in successor papers.Comment: 46 pages, LateX2E+AMSLaTe
Kinematic Quantities and Raychaudhuri Equations in a Universe
Based on some ideas emerged from the classical Kaluza-Klein theory, we
present a universe as a product bundle over the spacetime. This
enables us to introduce and study two categories of kinematic quantities
(expansions, shear, vorticity) in a universe. One category is related to
the fourth dimension (time), and the other one comes from the assumption of the
existence of the fifth dimension. The Raychaudhuri type equations that we
obtain in the paper, lead us to results on the evolution of both the
expansion and expansion in a universe.Comment: 27 page
- …