13,335 research outputs found
Nonperturbative Ambiguities and the Reality of Resurgent Transseries
In a wide range of quantum theoretical settings -- from quantum mechanics to
quantum field theory, from gauge theory to string theory -- singularities in
the complex Borel plane, usually associated to instantons or renormalons,
render perturbation theory ill-defined as they give rise to nonperturbative
ambiguities. These ambiguities are associated to choices of an integration
contour in the resummation of perturbation theory, along (singular) Stokes
directions in the complex Borel plane (rendering perturbative expansions
non-Borel summable along any Stokes line). More recently, it has been shown
that the proper framework to address these issues is that of resurgent analysis
and transseries. In this context, the cancelation of all nonperturbative
ambiguities is shown to be a consequence of choosing the transseries median
resummation as the appropriate family of unambiguous real solutions along the
coupling-constant real axis. While the median resummation is easily implemented
for one-parameter transseries, once one considers more general multi-parameter
transseries the procedure becomes highly dependent upon properly understanding
Stokes transitions in the complex Borel plane. In particular, all Stokes
coefficients must now be known in order to explicitly implement multi-parameter
median resummations. In the cases where quantum-theoretical physical
observables are described by resurgent functions and transseries, the methods
described herein show how one may cancel nonperturbative ambiguities, and
define these observables nonperturbatively starting out from perturbation
theory. Along the way, structural results concerning resurgent transseries are
also obtained.Comment: 62 pages, 4 figures; v2: corrected typos, added small discussion on
topological sectors, two new figure
Fully Bayesian Logistic Regression with Hyper-Lasso Priors for High-dimensional Feature Selection
High-dimensional feature selection arises in many areas of modern science.
For example, in genomic research we want to find the genes that can be used to
separate tissues of different classes (e.g. cancer and normal) from tens of
thousands of genes that are active (expressed) in certain tissue cells. To this
end, we wish to fit regression and classification models with a large number of
features (also called variables, predictors). In the past decade, penalized
likelihood methods for fitting regression models based on hyper-LASSO
penalization have received increasing attention in the literature. However,
fully Bayesian methods that use Markov chain Monte Carlo (MCMC) are still in
lack of development in the literature. In this paper we introduce an MCMC
(fully Bayesian) method for learning severely multi-modal posteriors of
logistic regression models based on hyper-LASSO priors (non-convex penalties).
Our MCMC algorithm uses Hamiltonian Monte Carlo in a restricted Gibbs sampling
framework; we call our method Bayesian logistic regression with hyper-LASSO
(BLRHL) priors. We have used simulation studies and real data analysis to
demonstrate the superior performance of hyper-LASSO priors, and to investigate
the issues of choosing heaviness and scale of hyper-LASSO priors.Comment: 33 pages. arXiv admin note: substantial text overlap with
arXiv:1308.469
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