5,761 research outputs found
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems
We address the numerical solution of infinite-dimensional inverse problems in
the framework of Bayesian inference. In the Part I companion to this paper
(arXiv.org:1308.1313), we considered the linearized infinite-dimensional
inverse problem. Here in Part II, we relax the linearization assumption and
consider the fully nonlinear infinite-dimensional inverse problem using a
Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of
sampling high-dimensional pdfs arising from Bayesian inverse problems governed
by PDEs, we build on the stochastic Newton MCMC method. This method exploits
problem structure by taking as a proposal density a local Gaussian
approximation of the posterior pdf, whose construction is made tractable by
invoking a low-rank approximation of its data misfit component of the Hessian.
Here we introduce an approximation of the stochastic Newton proposal in which
we compute the low-rank-based Hessian at just the MAP point, and then reuse
this Hessian at each MCMC step. We compare the performance of the proposed
method to the original stochastic Newton MCMC method and to an independence
sampler. The comparison of the three methods is conducted on a synthetic ice
sheet inverse problem. For this problem, the stochastic Newton MCMC method with
a MAP-based Hessian converges at least as rapidly as the original stochastic
Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian
at each step. On the other hand, it is more expensive per sample than the
independence sampler; however, its convergence is significantly more rapid, and
thus overall it is much cheaper. Finally, we present extensive analysis and
interpretation of the posterior distribution, and classify directions in
parameter space based on the extent to which they are informed by the prior or
the observations.Comment: 31 page
Tensor Regression with Applications in Neuroimaging Data Analysis
Classical regression methods treat covariates as a vector and estimate a
corresponding vector of regression coefficients. Modern applications in medical
imaging generate covariates of more complex form such as multidimensional
arrays (tensors). Traditional statistical and computational methods are proving
insufficient for analysis of these high-throughput data due to their ultrahigh
dimensionality as well as complex structure. In this article, we propose a new
family of tensor regression models that efficiently exploit the special
structure of tensor covariates. Under this framework, ultrahigh dimensionality
is reduced to a manageable level, resulting in efficient estimation and
prediction. A fast and highly scalable estimation algorithm is proposed for
maximum likelihood estimation and its associated asymptotic properties are
studied. Effectiveness of the new methods is demonstrated on both synthetic and
real MRI imaging data.Comment: 27 pages, 4 figure
Penalized Likelihood and Bayesian Function Selection in Regression Models
Challenging research in various fields has driven a wide range of
methodological advances in variable selection for regression models with
high-dimensional predictors. In comparison, selection of nonlinear functions in
models with additive predictors has been considered only more recently. Several
competing suggestions have been developed at about the same time and often do
not refer to each other. This article provides a state-of-the-art review on
function selection, focusing on penalized likelihood and Bayesian concepts,
relating various approaches to each other in a unified framework. In an
empirical comparison, also including boosting, we evaluate several methods
through applications to simulated and real data, thereby providing some
guidance on their performance in practice
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
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