7,370 research outputs found
Super-resolution Line Spectrum Estimation with Block Priors
We address the problem of super-resolution line spectrum estimation of an
undersampled signal with block prior information. The component frequencies of
the signal are assumed to take arbitrary continuous values in known frequency
blocks. We formulate a general semidefinite program to recover these
continuous-valued frequencies using theories of positive trigonometric
polynomials. The proposed semidefinite program achieves super-resolution
frequency recovery by taking advantage of known structures of frequency blocks.
Numerical experiments show great performance enhancements using our method.Comment: 7 pages, double colum
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
Detailed ultraviolet asymptotics for AdS scalar field perturbations
We present a range of methods suitable for accurate evaluation of the leading
asymptotics for integrals of products of Jacobi polynomials in limits when the
degrees of some or all polynomials inside the integral become large. The
structures in question have recently emerged in the context of effective
descriptions of small amplitude perturbations in anti-de Sitter (AdS)
spacetime. The limit of high degree polynomials corresponds in this situation
to effective interactions involving extreme short-wavelength modes, whose
dynamics is crucial for the turbulent instabilities that determine the ultimate
fate of small AdS perturbations. We explicitly apply the relevant asymptotic
techniques to the case of a self-interacting probe scalar field in AdS and
extract a detailed form of the leading large degree behavior, including closed
form analytic expressions for the numerical coefficients appearing in the
asymptotics.Comment: v2: 19 pages, expanded version accepted to JHE
From "Dirac combs" to Fourier-positivity
Motivated by various problems in physics and applied mathematics, we look for
constraints and properties of real Fourier-positive functions, i.e. with
positive Fourier transforms. Properties of the "Dirac comb" distribution and of
its tensor products in higher dimensions lead to Poisson resummation, allowing
for a useful approximation formula of a Fourier transform in terms of a limited
number of terms. A connection with the Bochner theorem on positive definiteness
of Fourier-positive functions is discussed. As a practical application, we find
simple and rapid analytic algorithms for checking Fourier-positivity in 1- and
(radial) 2-dimensions among a large variety of real positive functions. This
may provide a step towards a classification of positive positive-definite
functions.Comment: 17 pages, 14 eps figures (overall 8 figures in the text
Orthogonal sets of data windows constructed from trigonometric polynomials
Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated
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