14,145 research outputs found
Part I. The Cosmological Vacuum from a Topological Perspective
This article examines how the physical presence of field energy and
particulate matter can be interpreted in terms of the topological properties of
space-time. The theory is developed in terms of vector and matrix equations of
exterior differential systems, which are not constrained by tensor
diffeomorphic equivalences. The first postulate defines the field properties (a
vector space continuum) of the Cosmological Vacuum in terms of matrices of
basis functions that map exact differentials into neighborhoods of exterior
differential 1-forms (potentials). The second postulate requires that the field
equations must satisfy the First Law of Thermodynamics dynamically created in
terms of the Lie differential with respect to a process direction field acting
on the exterior differential forms that encode the thermodynamic system. The
vector space of infinitesimals need not be global and its compliment is used to
define particle properties as topological defects embedded in the field vector
space. The potentials, as exterior differential 1-forms, are not (necessarily)
uniquely integrable: the fibers can be twisted, leading to possible Chiral
matrix arrays of certain 3-forms defined as Topological Torsion and Topological
Spin. A significant result demonstrates how the coefficients of Affine Torsion
are related to the concept of Field excitations (mass and charge); another
demonstrates how thermodynamic evolution can describe the emergence of
topological defects in the physical vacuum.Comment: 70 pages, 5 figure
Learning to Approximate a Bregman Divergence
Bregman divergences generalize measures such as the squared Euclidean
distance and the KL divergence, and arise throughout many areas of machine
learning. In this paper, we focus on the problem of approximating an arbitrary
Bregman divergence from supervision, and we provide a well-principled approach
to analyzing such approximations. We develop a formulation and algorithm for
learning arbitrary Bregman divergences based on approximating their underlying
convex generating function via a piecewise linear function. We provide
theoretical approximation bounds using our parameterization and show that the
generalization error for metric learning using our framework
matches the known generalization error in the strictly less general Mahalanobis
metric learning setting. We further demonstrate empirically that our method
performs well in comparison to existing metric learning methods, particularly
for clustering and ranking problems.Comment: 19 pages, 4 figure
High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of and (div) methods
The accurate numerical simulation of turbulent incompressible flows is a
challenging topic in computational fluid dynamics. For discretisation methods
to be robust in the under-resolved regime, mass conservation as well as energy
stability are key ingredients to obtain robust and accurate discretisations.
Recently, two approaches have been proposed in the context of high-order
discontinuous Galerkin (DG) discretisations that address these aspects
differently. On the one hand, standard -based DG discretisations enforce
mass conservation and energy stability weakly by the use of additional
stabilisation terms. On the other hand, pointwise divergence-free
-conforming approaches ensure exact mass conservation
and energy stability by the use of tailored finite element function spaces. The
present work raises the question whether and to which extent these two
approaches are equivalent when applied to under-resolved turbulent flows. This
comparative study highlights similarities and differences of these two
approaches. The numerical results emphasise that both discretisation strategies
are promising for under-resolved simulations of turbulent flows due to their
inherent dissipation mechanisms.Comment: 24 pages, 13 figure
On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression
In this paper, we consider the nonparametric regression problem with
multivariate predictors. We provide a characterization of the degrees of
freedom and divergence for estimators of the unknown regression function, which
are obtained as outputs of linearly constrained quadratic optimization
procedures, namely, minimizers of the least squares criterion with linear
constraints and/or quadratic penalties. As special cases of our results, we
derive explicit expressions for the degrees of freedom in many nonparametric
regression problems, e.g., bounded isotonic regression, multivariate
(penalized) convex regression, and additive total variation regularization. Our
theory also yields, as special cases, known results on the degrees of freedom
of many well-studied estimators in the statistics literature, such as ridge
regression, Lasso and generalized Lasso. Our results can be readily used to
choose the tuning parameter(s) involved in the estimation procedure by
minimizing the Stein's unbiased risk estimate. As a by-product of our analysis
we derive an interesting connection between bounded isotonic regression and
isotonic regression on a general partially ordered set, which is of independent
interest.Comment: 72 pages, 7 figures, Journal of the American Statistical Association
(Theory and Methods), 201
A data driven equivariant approach to constrained Gaussian mixture modeling
Maximum likelihood estimation of Gaussian mixture models with different
class-specific covariance matrices is known to be problematic. This is due to
the unboundedness of the likelihood, together with the presence of spurious
maximizers. Existing methods to bypass this obstacle are based on the fact that
unboundedness is avoided if the eigenvalues of the covariance matrices are
bounded away from zero. This can be done imposing some constraints on the
covariance matrices, i.e. by incorporating a priori information on the
covariance structure of the mixture components. The present work introduces a
constrained equivariant approach, where the class conditional covariance
matrices are shrunk towards a pre-specified matrix Psi. Data-driven choices of
the matrix Psi, when a priori information is not available, and the optimal
amount of shrinkage are investigated. The effectiveness of the proposal is
evaluated on the basis of a simulation study and an empirical example
The Price equation program: simple invariances unify population dynamics, thermodynamics, probability, information and inference
The fundamental equations of various disciplines often seem to share the same
basic structure. Natural selection increases information in the same way that
Bayesian updating increases information. Thermodynamics and the forms of common
probability distributions express maximum increase in entropy, which appears
mathematically as loss of information. Physical mechanics follows paths of
change that maximize Fisher information. The information expressions typically
have analogous interpretations as the Newtonian balance between force and
acceleration, representing a partition between direct causes of change and
opposing changes in the frame of reference. This web of vague analogies hints
at a deeper common mathematical structure. I suggest that the Price equation
expresses that underlying universal structure. The abstract Price equation
describes dynamics as the change between two sets. One component of dynamics
expresses the change in the frequency of things, holding constant the values
associated with things. The other component of dynamics expresses the change in
the values of things, holding constant the frequency of things. The separation
of frequency from value generalizes Shannon's separation of the frequency of
symbols from the meaning of symbols in information theory. The Price equation's
generalized separation of frequency and value reveals a few simple invariances
that define universal geometric aspects of change. For example, the
conservation of total frequency, although a trivial invariance by itself,
creates a powerful constraint on the geometry of change. That constraint plus a
few others seem to explain the common structural forms of the equations in
different disciplines. From that abstract perspective, interpretations such as
selection, information, entropy, force, acceleration, and physical work arise
from the same underlying geometry expressed by the Price equation.Comment: Version 3: added figure illustrating geometry; added table of symbols
and two tables summarizing mathematical relations; this version accepted for
publication in Entrop
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