6,242 research outputs found
Choosing the Right Spatial Weighting Matrix in a Quantile Regression Model
This paper proposes computationally tractable methods for selecting the appropriate spatial weighting matrix in the context of a spatial quantile regression model. This selection is a notoriously difficult problem even in linear spatial models and is even more difficult in a quantile regression setup. The proposal is illustrated by an empirical example and manages to produce tractable models. One important feature of the proposed methodology is that by allowing different degrees and forms of spatial dependence across quantiles it further relaxes the usual quantile restriction attributable to the linear quantile regression. In this way we can obtain a more robust, with regard to potential functional misspecification, model, but nevertheless preserve the parametric rate of convergence and the established inferential apparatus associated with the linear quantile regression approach
Sparse image reconstruction on the sphere: analysis and synthesis
We develop techniques to solve ill-posed inverse problems on the sphere by
sparse regularisation, exploiting sparsity in both axisymmetric and directional
scale-discretised wavelet space. Denoising, inpainting, and deconvolution
problems, and combinations thereof, are considered as examples. Inverse
problems are solved in both the analysis and synthesis settings, with a number
of different sampling schemes. The most effective approach is that with the
most restricted solution-space, which depends on the interplay between the
adopted sampling scheme, the selection of the analysis/synthesis problem, and
any weighting of the l1 norm appearing in the regularisation problem. More
efficient sampling schemes on the sphere improve reconstruction fidelity by
restricting the solution-space and also by improving sparsity in wavelet space.
We apply the technique to denoise Planck 353 GHz observations, improving the
ability to extract the structure of Galactic dust emission, which is important
for studying Galactic magnetism.Comment: 11 pages, 6 Figure
Robust sparse image reconstruction of radio interferometric observations with purify
Next-generation radio interferometers, such as the Square Kilometre Array
(SKA), will revolutionise our understanding of the universe through their
unprecedented sensitivity and resolution. However, to realise these goals
significant challenges in image and data processing need to be overcome. The
standard methods in radio interferometry for reconstructing images, such as
CLEAN, have served the community well over the last few decades and have
survived largely because they are pragmatic. However, they produce
reconstructed inter\-ferometric images that are limited in quality and
scalability for big data. In this work we apply and evaluate alternative
interferometric reconstruction methods that make use of state-of-the-art sparse
image reconstruction algorithms motivated by compressive sensing, which have
been implemented in the PURIFY software package. In particular, we implement
and apply the proximal alternating direction method of multipliers (P-ADMM)
algorithm presented in a recent article. First, we assess the impact of the
interpolation kernel used to perform gridding and degridding on sparse image
reconstruction. We find that the Kaiser-Bessel interpolation kernel performs as
well as prolate spheroidal wave functions, while providing a computational
saving and an analytic form. Second, we apply PURIFY to real interferometric
observations from the Very Large Array (VLA) and the Australia Telescope
Compact Array (ATCA) and find images recovered by PURIFY are higher quality
than those recovered by CLEAN. Third, we discuss how PURIFY reconstructions
exhibit additional advantages over those recovered by CLEAN. The latest version
of PURIFY, with developments presented in this work, is made publicly
available.Comment: 22 pages, 10 figures, PURIFY code available at
http://basp-group.github.io/purif
A Framework for Directional and Higher-Order Reconstruction in Photoacoustic Tomography
Photoacoustic tomography is a hybrid imaging technique that combines high
optical tissue contrast with high ultrasound resolution. Direct reconstruction
methods such as filtered backprojection, time reversal and least squares suffer
from curved line artefacts and blurring, especially in case of limited angles
or strong noise. In recent years, there has been great interest in regularised
iterative methods. These methods employ prior knowledge on the image to provide
higher quality reconstructions. However, easy comparisons between regularisers
and their properties are limited, since many tomography implementations heavily
rely on the specific regulariser chosen. To overcome this bottleneck, we
present a modular reconstruction framework for photoacoustic tomography. It
enables easy comparisons between regularisers with different properties, e.g.
nonlinear, higher-order or directional. We solve the underlying minimisation
problem with an efficient first-order primal-dual algorithm. Convergence rates
are optimised by choosing an operator dependent preconditioning strategy. Our
reconstruction methods are tested on challenging 2D synthetic and experimental
data sets. They outperform direct reconstruction approaches for strong noise
levels and limited angle measurements, offering immediate benefits in terms of
acquisition time and quality. This work provides a basic platform for the
investigation of future advanced regularisation methods in photoacoustic
tomography.Comment: submitted to "Physics in Medicine and Biology". Changes from v1 to
v2: regularisation with directional wavelet has been added; new experimental
tests have been include
Online Matrix Completion Through Nuclear Norm Regularisation
It is the main goal of this paper to propose a novel method to perform matrix
completion on-line. Motivated by a wide variety of applications, ranging from
the design of recommender systems to sensor network localization through
seismic data reconstruction, we consider the matrix completion problem when
entries of the matrix of interest are observed gradually. Precisely, we place
ourselves in the situation where the predictive rule should be refined
incrementally, rather than recomputed from scratch each time the sample of
observed entries increases. The extension of existing matrix completion methods
to the sequential prediction context is indeed a major issue in the Big Data
era, and yet little addressed in the literature. The algorithm promoted in this
article builds upon the Soft Impute approach introduced in Mazumder et al.
(2010). The major novelty essentially arises from the use of a randomised
technique for both computing and updating the Singular Value Decomposition
(SVD) involved in the algorithm. Though of disarming simplicity, the method
proposed turns out to be very efficient, while requiring reduced computations.
Several numerical experiments based on real datasets illustrating its
performance are displayed, together with preliminary results giving it a
theoretical basis.Comment: Corrected a typo in the affiliatio
Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization
We propose a new first-order optimisation algorithm to solve high-dimensional
non-smooth composite minimisation problems. Typical examples of such problems
have an objective that decomposes into a non-smooth empirical risk part and a
non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal
Mirror-Prox, leverages the Fenchel-type representation of one part of the
objective while handling the other part of the objective via linear
minimization over the domain. The algorithm stands in contrast with more
classical proximal gradient algorithms with smoothing, which require the
computation of proximal operators at each iteration and can therefore be
impractical for high-dimensional problems. We establish the theoretical
convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal
complexity bounds, i.e. , for the number of calls to linear
minimization oracle. We present promising experimental results showing the
interest of the approach in comparison to competing methods
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