37 research outputs found
Minimum mean square distance estimation of a subspace
We consider the problem of subspace estimation in a Bayesian setting. Since
we are operating in the Grassmann manifold, the usual approach which consists
of minimizing the mean square error (MSE) between the true subspace and its
estimate may not be adequate as the MSE is not the natural metric in
the Grassmann manifold. As an alternative, we propose to carry out subspace
estimation by minimizing the mean square distance (MSD) between and its
estimate, where the considered distance is a natural metric in the Grassmann
manifold, viz. the distance between the projection matrices. We show that the
resulting estimator is no longer the posterior mean of but entails
computing the principal eigenvectors of the posterior mean of .
Derivation of the MMSD estimator is carried out in a few illustrative examples
including a linear Gaussian model for the data and a Bingham or von Mises
Fisher prior distribution for . In all scenarios, posterior distributions
are derived and the MMSD estimator is obtained either analytically or
implemented via a Markov chain Monte Carlo simulation method. The method is
shown to provide accurate estimates even when the number of samples is lower
than the dimension of . An application to hyperspectral imagery is finally
investigated
A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric
We derive a numerical algorithm for evaluating the Riemannian logarithm on
the Stiefel manifold with respect to the canonical metric. In contrast to the
existing optimization-based approach, we work from a purely matrix-algebraic
perspective. Moreover, we prove that the algorithm converges locally and
exhibits a linear rate of convergence.Comment: 30 pages, 5 figures, Matlab cod
Parametric Regression on the Grassmannian
We address the problem of fitting parametric curves on the Grassmann manifold
for the purpose of intrinsic parametric regression. As customary in the
literature, we start from the energy minimization formulation of linear
least-squares in Euclidean spaces and generalize this concept to general
nonflat Riemannian manifolds, following an optimal-control point of view. We
then specialize this idea to the Grassmann manifold and demonstrate that it
yields a simple, extensible and easy-to-implement solution to the parametric
regression problem. In fact, it allows us to extend the basic geodesic model to
(1) a time-warped variant and (2) cubic splines. We demonstrate the utility of
the proposed solution on different vision problems, such as shape regression as
a function of age, traffic-speed estimation and crowd-counting from
surveillance video clips. Most notably, these problems can be conveniently
solved within the same framework without any specifically-tailored steps along
the processing pipeline.Comment: 14 pages, 11 figure
On the Whitney distortion extension problem for and and its applications to interpolation and alignment of data in
Let , open. In this paper we provide a sharp
solution to the following Whitney distortion extension problems: (a) Let
be a map. If is compact (with some
geometry) and the restriction of to is an almost isometry with small
distortion, how to decide when there exists a one-to-one and
onto almost isometry with small distortion
which agrees with in a neighborhood of and a Euclidean motion
away from . (b) Let
be map. If is compact (with some geometry) and the
restriction of to is an almost isometry with small distortion, how
to decide when there exists a one-to-one and onto
almost isometry with small distortion which
agrees with in a neighborhood of and a Euclidean motion away from . Our results complement those of [14,15,20]
where there, is a finite set. In this case, the problem above is also a
problem of interpolation and alignment of data in .Comment: This is part three of four papers with C. Fefferman (arXiv:1411.2451,
arXiv:1411.2468, involve-v5-n2-p03-s.pdf) dealing with the problem of Whitney
type extensions of distortions from certain compact sets to distorted diffeomorphisms on $\Bbb R^n