4,160 research outputs found
A Risk Comparison of Ordinary Least Squares vs Ridge Regression
We compare the risk of ridge regression to a simple variant of ordinary least
squares, in which one simply projects the data onto a finite dimensional
subspace (as specified by a Principal Component Analysis) and then performs an
ordinary (un-regularized) least squares regression in this subspace. This note
shows that the risk of this ordinary least squares method is within a constant
factor (namely 4) of the risk of ridge regression.Comment: Appearing in JMLR 14, June 201
Transition from phase to generalized synchronization in time-delay systems
The notion of phase synchronization in time-delay systems, exhibiting highly
non-phase-coherent attractors, has not been realized yet even though it has
been well studied in chaotic dynamical systems without delay. We report the
identification of phase synchronization in coupled nonidentical piece-wise
linear and in coupled Mackey-Glass time-delay systems with highly
non-phase-coherent regimes. We show that there is a transition from
non-synchronized behavior to phase and then to generalized synchronization as a
function of coupling strength. We have introduced a transformation to capture
the phase of the non-phase coherent attractors, which works equally well for
both the time-delay systems. The instantaneous phases of the above coupled
systems calculated from the transformed attractors satisfy both the phase and
mean frequency locking conditions. These transitions are also characterized in
terms of recurrence based indices, namely generalized autocorrelation function
, correlation of probability of recurrence (CPR), joint probability of
recurrence (JPR) and similarity of probability of recurrence (SPR). We have
quantified the different synchronization regimes in terms of these indices. The
existence of phase synchronization is also characterized by typical transitions
in the Lyapunov exponents of the coupled time-delay systems.Comment: Accepted for publication in CHAO
Seismic modeling using the frozen Gaussian approximation
We adopt the frozen Gaussian approximation (FGA) for modeling seismic waves.
The method belongs to the category of ray-based beam methods. It decomposes
seismic wavefield into a set of Gaussian functions and propagates these
Gaussian functions along appropriate ray paths. As opposed to the classic
Gaussian-beam method, FGA keeps the Gaussians frozen (at a fixed width) during
the propagation process and adjusts their amplitudes to produce an accurate
approximation after summation. We perform the initial decomposition of seismic
data using a fast version of the Fourier-Bros-Iagolnitzer (FBI) transform and
propagate the frozen Gaussian beams numerically using ray tracing. A test using
a smoothed Marmousi model confirms the validity of FGA for accurate modeling of
seismic wavefields.Comment: 5 pages, 8 figure
High-order harmonic generation from inhomogeneous fields
We present theoretical studies of high-order harmonic generation (HHG)
produced by non-homogeneous fields as resulting from the illumination of
plasmonic nanostructures with a short laser pulse. We show that both the
inhomogeneity of the local fields and the confinement of the electron movement
play an important role in the HHG process and lead to the generation of even
harmonics and a significantly increased cutoff, more pronounced for the longer
wavelengths cases studied. In order to understand and characterize the new HHG
features we employ two different approaches: the numerical solution of the time
dependent Schr\"odinger equation (TDSE) and the semiclassical approach known as
Strong Field Approximation (SFA). Both approaches predict comparable results
and show the new features, but using the semiclassical arguments behind the SFA
and time-frequency analysis tools, we are able to fully understand the reasons
of the cutoff extension.Comment: 25 pages, 12 figure
Polymer quantization, singularity resolution and the 1/r^2 potential
We present a polymer quantization of the -lambda/r^2 potential on the
positive real line and compute numerically the bound state eigenenergies in
terms of the dimensionless coupling constant lambda. The singularity at the
origin is handled in two ways: first, by regularizing the potential and
adopting either symmetric or antisymmetric boundary conditions; second, by
keeping the potential unregularized but allowing the singularity to be balanced
by an antisymmetric boundary condition. The results are compared to the
semiclassical limit of the polymer theory and to the conventional Schrodinger
quantization on L_2(R_+). The various quantization schemes are in excellent
agreement for the highly excited states but differ for the low-lying states,
and the polymer spectrum is bounded below even when the Schrodinger spectrum is
not. We find as expected that for the antisymmetric boundary condition the
regularization of the potential is redundant: the polymer quantum theory is
well defined even with the unregularized potential and the regularization of
the potential does not significantly affect the spectrum.Comment: 21 pages, LaTeX including 7 figures. v2: analytic bounds improved;
references adde
- …