1,157 research outputs found
Wiener filter reloaded: fast signal reconstruction without preconditioning
We present a high performance solution to the Wiener filtering problem via a
formulation that is dual to the recently developed messenger technique. This
new dual messenger algorithm, like its predecessor, efficiently calculates the
Wiener filter solution of large and complex data sets without preconditioning
and can account for inhomogeneous noise distributions and arbitrary mask
geometries. We demonstrate the capabilities of this scheme in signal
reconstruction by applying it on a simulated cosmic microwave background (CMB)
temperature data set. The performance of this new method is compared to that of
the standard messenger algorithm and the preconditioned conjugate gradient
(PCG) approach, using a series of well-known convergence diagnostics and their
processing times, for the particular problem under consideration. This variant
of the messenger algorithm matches the performance of the PCG method in terms
of the effectiveness of reconstruction of the input angular power spectrum and
converges smoothly to the final solution. The dual messenger algorithm
outperforms the standard messenger and PCG methods in terms of execution time,
as it runs to completion around 2 and 3-4 times faster than the respective
methods, for the specific problem considered.Comment: 13 pages, 10 figures. Accepted for publication in MNRAS main journa
Stochastic approximation of score functions for Gaussian processes
We discuss the statistical properties of a recently introduced unbiased
stochastic approximation to the score equations for maximum likelihood
calculation for Gaussian processes. Under certain conditions, including bounded
condition number of the covariance matrix, the approach achieves storage
and nearly computational effort per optimization step, where is the
number of data sites. Here, we prove that if the condition number of the
covariance matrix is bounded, then the approximate score equations are nearly
optimal in a well-defined sense. Therefore, not only is the approximation
efficient to compute, but it also has comparable statistical properties to the
exact maximum likelihood estimates. We discuss a modification of the stochastic
approximation in which design elements of the stochastic terms mimic patterns
from a factorial design. We prove these designs are always at least as
good as the unstructured design, and we demonstrate through simulation that
they can produce a substantial improvement over random designs. Our findings
are validated by numerical experiments on simulated data sets of up to 1
million observations. We apply the approach to fit a space-time model to over
80,000 observations of total column ozone contained in the latitude band
-N during April 2012.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS627 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficient Wiener filtering without preconditioning
We present a new approach to calculate the Wiener filter solution of general
data sets. It is trivial to implement, flexible, numerically absolutely stable,
and guaranteed to converge. Most importantly, it does not require an ingenious
choice of preconditioner to work well. The method is capable of taking into
account inhomogeneous noise distributions and arbitrary mask geometries. It
iteratively builds up the signal reconstruction by means of a messenger field,
introduced to mediate between the different preferred bases in which signal and
noise properties can be specified most conveniently. Using cosmic microwave
background (CMB) radiation data as a showcase, we demonstrate the capabilities
of our scheme by computing Wiener filtered WMAP7 temperature and polarization
maps at full resolution for the first time. We show how the algorithm can be
modified to synthesize fluctuation maps, which, combined with the Wiener filter
solution, result in unbiased constrained signal realizations, consistent with
the observations. The algorithm performs well even on simulated CMB maps with
Planck resolution and dynamic range.Comment: 5 pages, 2 figures. Submitted to Astronomy and Astrophysics. Replaced
to match published versio
Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization
We address the inverse problem of cosmic large-scale structure reconstruction
from a Bayesian perspective. For a linear data model, a number of known and
novel reconstruction schemes, which differ in terms of the underlying signal
prior, data likelihood, and numerical inverse extra-regularization schemes are
derived and classified. The Bayesian methodology presented in this paper tries
to unify and extend the following methods: Wiener-filtering, Tikhonov
regularization, Ridge regression, Maximum Entropy, and inverse regularization
techniques. The inverse techniques considered here are the asymptotic
regularization, the Jacobi, Steepest Descent, Newton-Raphson,
Landweber-Fridman, and both linear and non-linear Krylov methods based on
Fletcher-Reeves, Polak-Ribiere, and Hestenes-Stiefel Conjugate Gradients. The
structures of the up-to-date highest-performing algorithms are presented, based
on an operator scheme, which permits one to exploit the power of fast Fourier
transforms. Using such an implementation of the generalized Wiener-filter in
the novel ARGO-software package, the different numerical schemes are
benchmarked with 1-, 2-, and 3-dimensional problems including structured white
and Poissonian noise, data windowing and blurring effects. A novel numerical
Krylov scheme is shown to be superior in terms of performance and fidelity.
These fast inverse methods ultimately will enable the application of sampling
techniques to explore complex joint posterior distributions. We outline how the
space of the dark-matter density field, the peculiar velocity field, and the
power spectrum can jointly be investigated by a Gibbs-sampling process. Such a
method can be applied for the redshift distortions correction of the observed
galaxies and for time-reversal reconstructions of the initial density field.Comment: 40 pages, 11 figure
An error estimate of Gaussian Recursive Filter in 3Dvar problem
Computational kernel of the three-dimensional variational data assimilation
(3D-Var) problem is a linear system, generally solved by means of an iterative
method. The most costly part of each iterative step is a matrix-vector product
with a very large covariance matrix having Gaussian correlation structure. This
operation may be interpreted as a Gaussian convolution, that is a very
expensive numerical kernel. Recursive Filters (RFs) are a well known way to
approximate the Gaussian convolution and are intensively applied in the
meteorology, in the oceanography and in forecast models. In this paper, we deal
with an oceanographic 3D-Var data assimilation scheme, named OceanVar, where
the linear system is solved by using the Conjugate Gradient (GC) method by
replacing, at each step, the Gaussian convolution with RFs. Here we give
theoretical issues on the discrete convolution approximation with a first order
(1st-RF) and a third order (3rd-RF) recursive filters. Numerical experiments
confirm given error bounds and show the benefits, in terms of accuracy and
performance, of the 3-rd RF.Comment: 9 page
Preconditioning Kernel Matrices
The computational and storage complexity of kernel machines presents the
primary barrier to their scaling to large, modern, datasets. A common way to
tackle the scalability issue is to use the conjugate gradient algorithm, which
relieves the constraints on both storage (the kernel matrix need not be stored)
and computation (both stochastic gradients and parallelization can be used).
Even so, conjugate gradient is not without its own issues: the conditioning of
kernel matrices is often such that conjugate gradients will have poor
convergence in practice. Preconditioning is a common approach to alleviating
this issue. Here we propose preconditioned conjugate gradients for kernel
machines, and develop a broad range of preconditioners particularly useful for
kernel matrices. We describe a scalable approach to both solving kernel
machines and learning their hyperparameters. We show this approach is exact in
the limit of iterations and outperforms state-of-the-art approximations for a
given computational budget
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