283,818 research outputs found
WARP: Wavelets with adaptive recursive partitioning for multi-dimensional data
Effective identification of asymmetric and local features in images and other
data observed on multi-dimensional grids plays a critical role in a wide range
of applications including biomedical and natural image processing. Moreover,
the ever increasing amount of image data, in terms of both the resolution per
image and the number of images processed per application, requires algorithms
and methods for such applications to be computationally efficient. We develop a
new probabilistic framework for multi-dimensional data to overcome these
challenges through incorporating data adaptivity into discrete wavelet
transforms, thereby allowing them to adapt to the geometric structure of the
data while maintaining the linear computational scalability. By exploiting a
connection between the local directionality of wavelet transforms and recursive
dyadic partitioning on the grid points of the observation, we obtain the
desired adaptivity through adding to the traditional Bayesian wavelet
regression framework an additional layer of Bayesian modeling on the space of
recursive partitions over the grid points. We derive the corresponding
inference recipe in the form of a recursive representation of the exact
posterior, and develop a class of efficient recursive message passing
algorithms for achieving exact Bayesian inference with a computational
complexity linear in the resolution and sample size of the images. While our
framework is applicable to a range of problems including multi-dimensional
signal processing, compression, and structural learning, we illustrate its work
and evaluate its performance in the context of 2D and 3D image reconstruction
using real images from the ImageNet database. We also apply the framework to
analyze a data set from retinal optical coherence tomography
Gap Theorems for Locally Conformally Flat Manifolds
In this paper, we prove a gap result for a locally conformally flat complete
non-compact Riemannian manifold with bounded non-negative Ricci curvature and a
scalar curvature average condition. We show that if it has positive Green
function, then it is flat. This result is proved by setting up new global
Yamabe flow. Other extensions related to bounded positive solutions to a
schrodinger equation are also discussed.Comment: Accepted version in Journal of Differential Equatrio
Scalable Bayesian model averaging through local information propagation
We show that a probabilistic version of the classical forward-stepwise
variable inclusion procedure can serve as a general data-augmentation scheme
for model space distributions in (generalized) linear models. This latent
variable representation takes the form of a Markov process, thereby allowing
information propagation algorithms to be applied for sampling from model space
posteriors. In particular, we propose a sequential Monte Carlo method for
achieving effective unbiased Bayesian model averaging in high-dimensional
problems, utilizing proposal distributions constructed using local information
propagation. We illustrate our method---called LIPS for local information
propagation based sampling---through real and simulated examples with
dimensionality ranging from 15 to 1,000, and compare its performance in
estimating posterior inclusion probabilities and in out-of-sample prediction to
those of several other methods---namely, MCMC, BAS, iBMA, and LASSO. In
addition, we show that the latent variable representation can also serve as a
modeling tool for specifying model space priors that account for knowledge
regarding model complexity and conditional inclusion relationships
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