441 research outputs found
Extrinsic local regression on manifold-valued data
We propose an extrinsic regression framework for modeling data with manifold
valued responses and Euclidean predictors. Regression with manifold responses
has wide applications in shape analysis, neuroscience, medical imaging and many
other areas. Our approach embeds the manifold where the responses lie onto a
higher dimensional Euclidean space, obtains a local regression estimate in that
space, and then projects this estimate back onto the image of the manifold.
Outside the regression setting both intrinsic and extrinsic approaches have
been proposed for modeling i.i.d manifold-valued data. However, to our
knowledge our work is the first to take an extrinsic approach to the regression
problem. The proposed extrinsic regression framework is general,
computationally efficient and theoretically appealing. Asymptotic distributions
and convergence rates of the extrinsic regression estimates are derived and a
large class of examples are considered indicating the wide applicability of our
approach
Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction
We reframe linear dimensionality reduction as a problem of Bayesian inference
on matrix manifolds. This natural paradigm extends the Bayesian framework to
dimensionality reduction tasks in higher dimensions with simpler models at
greater speeds. Here an orthogonal basis is treated as a single point on a
manifold and is associated with a linear subspace on which observations vary
maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds
for various dimensionality reduction problems, explore the connection between
the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the
Grassmannian for the first time. We delineate in which situations either
manifold should be considered. Further, matrix manifold models are used to
yield scientific insight in the context of cognitive neuroscience, and we
conclude that our methods are suitable for basic inference as well as accurate
prediction.Comment: All datasets and computer programs are publicly available at
http://www.ics.uci.edu/~babaks/Site/Codes.htm
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
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