1,872 research outputs found
Continuous Shearlet Tight Frames
Based on the shearlet transform we present a general construction of
continuous tight frames for from any sufficiently smooth
function with anisotropic moments. This includes for example compactly
supported systems, piecewise polynomial systems, or both. From our earlier
results it follows that these systems enjoy the same desirable approximation
properties for directional data as the previous bandlimited and very specific
constructions due to Kutyniok and Labate. We also show that the representation
formulas we derive are in a sense optimal for the shearlet transform
Parabolic Molecules
Anisotropic decompositions using representation systems based on parabolic
scaling such as curvelets or shearlets have recently attracted significantly
increased attention due to the fact that they were shown to provide optimally
sparse approximations of functions exhibiting singularities on lower
dimensional embedded manifolds. The literature now contains various direct
proofs of this fact and of related sparse approximation results. However, it
seems quite cumbersome to prove such a canon of results for each system
separately, while many of the systems exhibit certain similarities.
In this paper, with the introduction of the notion of {\em parabolic
molecules}, we aim to provide a comprehensive framework which includes
customarily employed representation systems based on parabolic scaling such as
curvelets and shearlets. It is shown that pairs of parabolic molecules have the
fundamental property to be almost orthogonal in a particular sense. This result
is then applied to analyze parabolic molecules with respect to their ability to
sparsely approximate data governed by anisotropic features. For this, the
concept of {\em sparsity equivalence} is introduced which is shown to allow the
identification of a large class of parabolic molecules providing the same
sparse approximation results as curvelets and shearlets. Finally, as another
application, smoothness spaces associated with parabolic molecules are
introduced providing a general theoretical approach which even leads to novel
results for, for instance, compactly supported shearlets
Definability and stability of multiscale decompositions for manifold-valued data
We discuss multiscale representations of discrete manifold-valued data. As it
turns out that we cannot expect general manifold-analogues of biorthogonal
wavelets to possess perfect reconstruction, we focus our attention on those
constructions which are based on upscaling operators which are either
interpolating or midpoint-interpolating. For definable multiscale
decompositions we obtain a stability result
The surprising influence of late charged current weak interactions on Big Bang Nucleosynthesis
The weak interaction charged current processes (, , ) interconvert neutrons and protons in the early universe and
have significant influence on Big Bang Nucleosynthesis (BBN) light-element
abundance yields, particulary that for . We demonstrate that the
influence of these processes is still significant even when they operate well
below temperatures usually invoked for "weak freeze-out,"
and in fact down nearly into the alpha-particle formation epoch (). This physics is correctly captured in commonly used BBN
codes, though this late-time, low-temperature persistent effect of the
isospin-changing weak processes, and the sensitivity of the associated rates to
lepton energy distribution functions and blocking factors are not widely
appreciated. We quantify this late-time influence by analyzing weak interaction
rate dependence on the neutron lifetime, lepton energy distribution functions,
entropy, the proton-neutron mass difference, and Hubble expansion rate. The
effects we point out here render BBN a keen probe of any beyond-standard-model
physics that alters lepton number/energy distributions, even subtly, in epochs
of the early universe all the way down to near .Comment: 27 pages, 8 figure
Insights into neutrino decoupling gleaned from considerations of the role of electron mass
We present calculations showing how electron rest mass influences entropy
flow, neutrino decoupling, and Big Bang Nucleosynthesis (BBN) in the early
universe. To elucidate this physics and especially the sensitivity of BBN and
related epochs to electron mass, we consider a parameter space of rest mass
values larger and smaller than the accepted vacuum value. Electromagnetic
equilibrium, coupled with the high entropy of the early universe, guarantees
that significant numbers of electron-positron pairs are present, and dominate
over the number of ionization electrons to temperatures much lower than the
vacuum electron rest mass. Scattering between the electrons-positrons and the
neutrinos largely controls the flow of entropy from the plasma into the
neutrino seas. Moreover, the number density of electron-positron-pair targets
can be exponentially sensitive to the effective in-medium electron mass. This
entropy flow influences the phasing of scale factor and temperature, the
charged current weak-interaction-determined neutron-to-proton ratio, and the
spectral distortions in the relic neutrino energy spectra. Our calculations
show the sensitivity of the physics of this epoch to three separate effects:
finite electron mass, finite-temperature quantum electrodynamic (QED) effects
on the plasma equation of state, and Boltzmann neutrino energy transport. The
ratio of neutrino to plasma component energy scales manifests in Cosmic
Microwave Background (CMB) observables, namely the baryon density and the
radiation energy density, along with the primordial helium and deuterium
abundances. Our results demonstrate how the treatment of in-medium electron
mass (i.e., QED effects) could translate into an important source of
uncertainty in extracting neutrino and beyond-standard-model physics limits
from future high-precision CMB data.Comment: 32 pages, 8 figures, 1 table. Version accepted by Nuclear Physics
How degenerate is the parametrization of neural networks with the ReLU activation function?
Neural network training is usually accomplished by solving a non-convex
optimization problem using stochastic gradient descent. Although one optimizes
over the networks parameters, the main loss function generally only depends on
the realization of the neural network, i.e. the function it computes. Studying
the optimization problem over the space of realizations opens up new ways to
understand neural network training. In particular, usual loss functions like
mean squared error and categorical cross entropy are convex on spaces of neural
network realizations, which themselves are non-convex. Approximation
capabilities of neural networks can be used to deal with the latter
non-convexity, which allows us to establish that for sufficiently large
networks local minima of a regularized optimization problem on the realization
space are almost optimal. Note, however, that each realization has many
different, possibly degenerate, parametrizations. In particular, a local
minimum in the parametrization space needs not correspond to a local minimum in
the realization space. To establish such a connection, inverse stability of the
realization map is required, meaning that proximity of realizations must imply
proximity of corresponding parametrizations. We present pathologies which
prevent inverse stability in general, and, for shallow networks, proceed to
establish a restricted space of parametrizations on which we have inverse
stability w.r.t. to a Sobolev norm. Furthermore, we show that by optimizing
over such restricted sets, it is still possible to learn any function which can
be learned by optimization over unrestricted sets.Comment: Accepted at NeurIPS 201
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