45,467 research outputs found
On the accuracy of solving confluent Prony systems
In this paper we consider several nonlinear systems of algebraic equations
which can be called "Prony-type". These systems arise in various reconstruction
problems in several branches of theoretical and applied mathematics, such as
frequency estimation and nonlinear Fourier inversion. Consequently, the
question of stability of solution with respect to errors in the right-hand side
becomes critical for the success of any particular application. We investigate
the question of "maximal possible accuracy" of solving Prony-type systems,
putting stress on the "local" behavior which approximates situations with low
absolute measurement error. The accuracy estimates are formulated in very
simple geometric terms, shedding some light on the structure of the problem.
Numerical tests suggest that "global" solution techniques such as Prony's
algorithm and ESPRIT method are suboptimal when compared to this theoretical
"best local" behavior
A Deep Representation for Invariance And Music Classification
Representations in the auditory cortex might be based on mechanisms similar
to the visual ventral stream; modules for building invariance to
transformations and multiple layers for compositionality and selectivity. In
this paper we propose the use of such computational modules for extracting
invariant and discriminative audio representations. Building on a theory of
invariance in hierarchical architectures, we propose a novel, mid-level
representation for acoustical signals, using the empirical distributions of
projections on a set of templates and their transformations. Under the
assumption that, by construction, this dictionary of templates is composed from
similar classes, and samples the orbit of variance-inducing signal
transformations (such as shift and scale), the resulting signature is
theoretically guaranteed to be unique, invariant to transformations and stable
to deformations. Modules of projection and pooling can then constitute layers
of deep networks, for learning composite representations. We present the main
theoretical and computational aspects of a framework for unsupervised learning
of invariant audio representations, empirically evaluated on music genre
classification.Comment: 5 pages, CBMM Memo No. 002, (to appear) IEEE 2014 International
Conference on Acoustics, Speech, and Signal Processing (ICASSP 2014
Spectral functions and time evolution from the Chebyshev recursion
We link linear prediction of Chebyshev and Fourier expansions to analytic
continuation. We push the resolution in the Chebyshev-based computation of
many-body spectral functions to a much higher precision by deriving a
modified Chebyshev series expansion that allows to reduce the expansion order
by a factor . We show that in a certain limit the Chebyshev
technique becomes equivalent to computing spectral functions via time evolution
and subsequent Fourier transform. This introduces a novel recursive time
evolution algorithm that instead of the group operator only involves
the action of the generator . For quantum impurity problems, we introduce an
adapted discretization scheme for the bath spectral function. We discuss the
relevance of these results for matrix product state (MPS) based DMRG-type
algorithms, and their use within dynamical mean-field theory (DMFT). We present
strong evidence that the Chebyshev recursion extracts less spectral information
from than time evolution algorithms when fixing a given amount of created
entanglement.Comment: 12 pages + 6 pages appendix, 11 figure
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
Molecular Dipolar Crystals as High Fidelity Quantum Memory for Hybrid Quantum Computing
We study collective excitations of rotational and spin states of an ensemble
of polar molecules, which are prepared in a dipolar crystalline phase, as a
candidate for a high fidelity quantum memory. While dipolar crystals are formed
in the high density limit of cold clouds of polar molecules under 1D and 2D
trapping conditions, the crystalline structure protects the molecular qubits
from detrimental effects of short range collisions. We calculate the lifetime
of the quantum memory by identifying the dominant decoherence mechanisms, and
estimate their effects on gate operations, when a molecular ensemble qubit is
transferred to a superconducting strip line cavity (circuit QED). In the case
rotational excitations coupled by dipole-dipole interactions we identify
phonons as the main limitation of the life time of qubits. We study specific
setups and conditions, where the coupling to the phonon modes is minimized.
Detailed results are presented for a 1D dipolar chain
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