482 research outputs found
An efficient shooting algorithm for Evans function calculations in large systems
In Evans function computations of the spectra of asymptotically
constant-coefficient linear operators, a basic issue is the efficient and
numerically stable computation of subspaces evolving according to the
associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be
obtained by representing subspaces as single exterior products
\cite{AS,Br.1,Br.2,BrZ,BDG}. For large systems, however, the dimension of the
exterior-product space quickly becomes prohibitive, growing as ,
where is the dimension of the system written as a first-order ODE and
(typically ) is the dimension of the subspace. We resolve this
difficulty by the introduction of a simple polar coordinate algorithm
representing ``pure'' (monomial) products as scalar multiples of orthonormal
bases, for which the angular equation is a numerically optimized version of the
continuous orthogonalization method of Drury--Davey \cite{Da,Dr} and the radial
equation is evaluable by quadrature. Notably, the polar-coordinate method
preserves the important property of analyticity with respect to parameters.Comment: 21 pp., two figure
Bayesian orthogonal component analysis for sparse representation
This paper addresses the problem of identifying a lower dimensional space
where observed data can be sparsely represented. This under-complete dictionary
learning task can be formulated as a blind separation problem of sparse sources
linearly mixed with an unknown orthogonal mixing matrix. This issue is
formulated in a Bayesian framework. First, the unknown sparse sources are
modeled as Bernoulli-Gaussian processes. To promote sparsity, a weighted
mixture of an atom at zero and a Gaussian distribution is proposed as prior
distribution for the unobserved sources. A non-informative prior distribution
defined on an appropriate Stiefel manifold is elected for the mixing matrix.
The Bayesian inference on the unknown parameters is conducted using a Markov
chain Monte Carlo (MCMC) method. A partially collapsed Gibbs sampler is
designed to generate samples asymptotically distributed according to the joint
posterior distribution of the unknown model parameters and hyperparameters.
These samples are then used to approximate the joint maximum a posteriori
estimator of the sources and mixing matrix. Simulations conducted on synthetic
data are reported to illustrate the performance of the method for recovering
sparse representations. An application to sparse coding on under-complete
dictionary is finally investigated.Comment: Revised version. Accepted to IEEE Trans. Signal Processin
Space Frequency Codes from Spherical Codes
A new design method for high rate, fully diverse ('spherical') space
frequency codes for MIMO-OFDM systems is proposed, which works for arbitrary
numbers of antennas and subcarriers. The construction exploits a differential
geometric connection between spherical codes and space time codes. The former
are well studied e.g. in the context of optimal sequence design in CDMA
systems, while the latter serve as basic building blocks for space frequency
codes. In addition a decoding algorithm with moderate complexity is presented.
This is achieved by a lattice based construction of spherical codes, which
permits lattice decoding algorithms and thus offers a substantial reduction of
complexity.Comment: 5 pages. Final version for the 2005 IEEE International Symposium on
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