754 research outputs found
Gauge invariance in Loop Quantum Cosmology : Hamilton-Jacobi and Mukhanov-Sasaki equations for scalar perturbations
Gauge invariance of scalar perturbations is studied together with the
associated equations of motion. Extending methods developed in the framework of
hamiltonian General Relativity, the Hamilton-Jacobi equation is investigated
into the details in Loop Quantum Cosmology. The gauge-invariant observables are
built and their equations of motions are reviewed both in Hamiltonian and
Lagrangian approaches. This method is applied to scalar perturbations with
either holonomy or inverse-volume corrections.Comment: 16 page
Anomaly-free perturbations with inverse-volume and holonomy corrections in Loop Quantum Cosmology
This article addresses the issue of the closure of the algebra of constraints
for generic (cosmological) perturbations when taking into account
simultaneously the two main corrections of effective loop quantum cosmology,
namely the holonomy and the inverse-volume terms. Previous works on either the
holonomy or the inverse volume case are reviewed and generalized. In the
inverse-volume case, we point out new possibilities. An anomaly-free solution
including both corrections is found for perturbations, and the corresponding
equations of motion are derived.Comment: previous mistake corrected leading to new result
Primordial tensor power spectrum in holonomy corrected Omega-LQC
The holonomy correction is one of the main terms arising when implementing
loop quantum gravity ideas at an effective level in cosmology. The recent
construction of an anomaly free algebra has shown that the formalism used, up
to now, to derive the primordial spectrum of fluctuations was not correct. This
article aims at computing the tensor spectrum in a fully consistent way within
this deformed and closed algebra.Comment: 5 pages, 6 figures, accepted by Phys. Rev.
Stabilizing Training of Generative Adversarial Networks through Regularization
Deep generative models based on Generative Adversarial Networks (GANs) have
demonstrated impressive sample quality but in order to work they require a
careful choice of architecture, parameter initialization, and selection of
hyper-parameters. This fragility is in part due to a dimensional mismatch or
non-overlapping support between the model distribution and the data
distribution, causing their density ratio and the associated f-divergence to be
undefined. We overcome this fundamental limitation and propose a new
regularization approach with low computational cost that yields a stable GAN
training procedure. We demonstrate the effectiveness of this regularizer across
several architectures trained on common benchmark image generation tasks. Our
regularization turns GAN models into reliable building blocks for deep
learning
Inflation in loop quantum cosmology: Dynamics and spectrum of gravitational waves
Loop quantum cosmology provides an efficient framework to study the evolution
of the Universe beyond the classical Big Bang paradigm. Because of holonomy
corrections, the singularity is replaced by a "bounce". The dynamics of the
background is investigated into the details, as a function of the parameters of
the model. In particular, the conditions required for inflation to occur are
carefully considered and are shown to be generically met. The propagation of
gravitational waves is then investigated in this framework. By both numerical
and analytical approaches, the primordial tensor power spectrum is computed for
a wide range of parameters. Several interesting features could be
observationally probed.Comment: 11 pages, 14 figures. Matches version published in Phys. Rev.
Variance Reduced Stochastic Gradient Descent with Neighbors
Stochastic Gradient Descent (SGD) is a workhorse in machine learning, yet its
slow convergence can be a computational bottleneck. Variance reduction
techniques such as SAG, SVRG and SAGA have been proposed to overcome this
weakness, achieving linear convergence. However, these methods are either based
on computations of full gradients at pivot points, or on keeping per data point
corrections in memory. Therefore speed-ups relative to SGD may need a minimal
number of epochs in order to materialize. This paper investigates algorithms
that can exploit neighborhood structure in the training data to share and
re-use information about past stochastic gradients across data points, which
offers advantages in the transient optimization phase. As a side-product we
provide a unified convergence analysis for a family of variance reduction
algorithms, which we call memorization algorithms. We provide experimental
results supporting our theory.Comment: Appears in: Advances in Neural Information Processing Systems 28
(NIPS 2015). 13 page
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