8,119 research outputs found
Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver
The FEAST method for solving large sparse eigenproblems is equivalent to
subspace iteration with an approximate spectral projector and implicit
orthogonalization. This relation allows to characterize the convergence of this
method in terms of the error of a certain rational approximant to an indicator
function. We propose improved rational approximants leading to FEAST variants
with faster convergence, in particular, when using rational approximants based
on the work of Zolotarev. Numerical experiments demonstrate the possible
computational savings especially for pencils whose eigenvalues are not well
separated and when the dimension of the search space is only slightly larger
than the number of wanted eigenvalues. The new approach improves both
convergence robustness and load balancing when FEAST runs on multiple search
intervals in parallel.Comment: 22 pages, 8 figure
A joint time-invariant filtering approach to the linear Gaussian relay problem
In this paper, the linear Gaussian relay problem is considered. Under the
linear time-invariant (LTI) model the problem is formulated in the frequency
domain based on the Toeplitz distribution theorem. Under the further assumption
of realizable input spectra, the LTI Gaussian relay problem is converted to a
joint design problem of source and relay filters under two power constraints,
one at the source and the other at the relay, and a practical solution to this
problem is proposed based on the projected subgradient method. Numerical
results show that the proposed method yields a noticeable gain over the
instantaneous amplify-and-forward (AF) scheme in inter-symbol interference
(ISI) channels. Also, the optimality of the AF scheme within the class of
one-tap relay filters is established in flat-fading channels.Comment: 30 pages, 10 figure
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
A new family of high-resolution multivariate spectral estimators
In this paper, we extend the Beta divergence family to multivariate power
spectral densities. Similarly to the scalar case, we show that it smoothly
connects the multivariate Kullback-Leibler divergence with the multivariate
Itakura-Saito distance. We successively study a spectrum approximation problem,
based on the Beta divergence family, which is related to a multivariate
extension of the THREE spectral estimation technique. It is then possible to
characterize a family of solutions to the problem. An upper bound on the
complexity of these solutions will also be provided. Simulations suggest that
the most suitable solution of this family depends on the specific features
required from the estimation problem
Likelihood Analysis of Power Spectra and Generalized Moment Problems
We develop an approach to spectral estimation that has been advocated by
Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance
extension problem, by Enqvist and Karlsson. The aim is to determine the power
spectrum that is consistent with given moments and minimizes the relative
entropy between the probability law of the underlying Gaussian stochastic
process to that of a prior. The approach is analogous to the framework of
earlier work by Byrnes, Georgiou and Lindquist and can also be viewed as a
generalization of the classical work by Burg and Jaynes on the maximum entropy
method. In the present paper we present a new fast algorithm in the general
case (i.e., for general Gaussian priors) and show that for priors with a
specific structure the solution can be given in closed form.Comment: 17 pages, 4 figure
Convergence analysis of a family of robust Kalman filters based on the contraction principle
In this paper we analyze the convergence of a family of robust Kalman
filters. For each filter of this family the model uncertainty is tuned
according to the so called tolerance parameter. Assuming that the corresponding
state-space model is reachable and observable, we show that the corresponding
Riccati-like mapping is strictly contractive provided that the tolerance is
sufficiently small, accordingly the filter converges
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