10,640 research outputs found
Regularization Methods in Chiral Perturbation Theory
Chiral lagrangians describing the interactions of Goldstone bosons in a
theory possessing spontaneous symmetry breaking are effective,
non-renormalizable field theories in four dimensions. Yet, in a momentum
expansion one is able to extract definite, testable predictions from
perturbation theory. These techniques have yielded in recent years a wealth of
information on many problems where the physics of Goldstone bosons plays a
crucial role, but theoretical issues concerning chiral perturbation theory
remain, to this date, poorly treated in the literature. We present here a
rather comprehensive analysis of the regularization and renormalization
ambiguities appearing in chiral perturbation theory at the one loop level. We
discuss first on the relevance of dealing with tadpoles properly. We
demonstrate that Ward identities severely constrain the choice of regulators to
the point of enforcing unique, unambiguous results in chiral perturbation
theory at the one-loop level for any observable which is renormalization-group
invariant. We comment on the physical implications of these results and on
several possible regulating methods that may be of use for some applications.Comment: 37 pages, 5 figs. not included (available upon request), LaTeX,
PREPRINT UB-ECM-PF 93/1
Regularization Methods for Nuclear Lattice Effective Field Theory
We investigate Nuclear Lattice Effective Field Theory for the two-body system
for several lattice spacings at lowest order in the pionless as well as in the
pionful theory. We discuss issues of regularizations and predictions for the
effective range expansion. In the pionless case, a simple Gaussian smearing
allows to demonstrate lattice spacing independence over a wide range of lattice
spacings. We show that regularization methods known from the continuum
formulation are necessary as well as feasible for the pionful approach.Comment: 7 pp, 2 figs, to appear in Physics Letters
An analytic comparison of regularization methods for Gaussian Processes
Gaussian Processes (GPs) are a popular approach to predict the output of a
parameterized experiment. They have many applications in the field of Computer
Experiments, in particular to perform sensitivity analysis, adaptive design of
experiments and global optimization. Nearly all of the applications of GPs
require the inversion of a covariance matrix that, in practice, is often
ill-conditioned. Regularization methodologies are then employed with
consequences on the GPs that need to be better understood.The two principal
methods to deal with ill-conditioned covariance matrices are i) pseudoinverse
and ii) adding a positive constant to the diagonal (the so-called nugget
regularization).The first part of this paper provides an algebraic comparison
of PI and nugget regularizations. Redundant points, responsible for covariance
matrix singularity, are defined. It is proven that pseudoinverse
regularization, contrarily to nugget regularization, averages the output values
and makes the variance zero at redundant points. However, pseudoinverse and
nugget regularizations become equivalent as the nugget value vanishes. A
measure for data-model discrepancy is proposed which serves for choosing a
regularization technique.In the second part of the paper, a distribution-wise
GP is introduced that interpolates Gaussian distributions instead of data
points. Distribution-wise GP can be seen as an improved regularization method
for GPs
Higher covariant derivative regulators and non-multiplicative renormalization
The renormalization algorithm based on regularization methods with two
regulators is analyzed by means of explicit computations. We show in particular
that regularization by higher covariant derivative terms can be complemented
with dimensional regularization to obtain a consistent renormalized
4-dimensional Yang-Mills theory at the one-loop level. This shows that hybrid
regularization methods can be applied not only to finite theories, like \eg\
Chern-Simons, but also to divergent theories.Comment: 12 pages, phyzzx, no figure
Model Selection for High Dimensional Quadratic Regression via Regularization
Quadratic regression (QR) models naturally extend linear models by
considering interaction effects between the covariates. To conduct model
selection in QR, it is important to maintain the hierarchical model structure
between main effects and interaction effects. Existing regularization methods
generally achieve this goal by solving complex optimization problems, which
usually demands high computational cost and hence are not feasible for high
dimensional data. This paper focuses on scalable regularization methods for
model selection in high dimensional QR. We first consider two-stage
regularization methods and establish theoretical properties of the two-stage
LASSO. Then, a new regularization method, called Regularization Algorithm under
Marginality Principle (RAMP), is proposed to compute a hierarchy-preserving
regularization solution path efficiently. Both methods are further extended to
solve generalized QR models. Numerical results are also shown to demonstrate
performance of the methods.Comment: 37 pages, 1 figure with supplementary materia
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