8,871 research outputs found
Out-of-sample generalizations for supervised manifold learning for classification
Supervised manifold learning methods for data classification map data samples
residing in a high-dimensional ambient space to a lower-dimensional domain in a
structure-preserving way, while enhancing the separation between different
classes in the learned embedding. Most nonlinear supervised manifold learning
methods compute the embedding of the manifolds only at the initially available
training points, while the generalization of the embedding to novel points,
known as the out-of-sample extension problem in manifold learning, becomes
especially important in classification applications. In this work, we propose a
semi-supervised method for building an interpolation function that provides an
out-of-sample extension for general supervised manifold learning algorithms
studied in the context of classification. The proposed algorithm computes a
radial basis function (RBF) interpolator that minimizes an objective function
consisting of the total embedding error of unlabeled test samples, defined as
their distance to the embeddings of the manifolds of their own class, as well
as a regularization term that controls the smoothness of the interpolation
function in a direction-dependent way. The class labels of test data and the
interpolation function parameters are estimated jointly with a progressive
procedure. Experimental results on face and object images demonstrate the
potential of the proposed out-of-sample extension algorithm for the
classification of manifold-modeled data sets
Regression on manifolds: Estimation of the exterior derivative
Collinearity and near-collinearity of predictors cause difficulties when
doing regression. In these cases, variable selection becomes untenable because
of mathematical issues concerning the existence and numerical stability of the
regression coefficients, and interpretation of the coefficients is ambiguous
because gradients are not defined. Using a differential geometric
interpretation, in which the regression coefficients are interpreted as
estimates of the exterior derivative of a function, we develop a new method to
do regression in the presence of collinearities. Our regularization scheme can
improve estimation error, and it can be easily modified to include lasso-type
regularization. These estimators also have simple extensions to the "large ,
small " context.Comment: Published in at http://dx.doi.org/10.1214/10-AOS823 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optical Flow on Moving Manifolds
Optical flow is a powerful tool for the study and analysis of motion in a
sequence of images. In this article we study a Horn-Schunck type
spatio-temporal regularization functional for image sequences that have a
non-Euclidean, time varying image domain. To that end we construct a Riemannian
metric that describes the deformation and structure of this evolving surface.
The resulting functional can be seen as natural geometric generalization of
previous work by Weickert and Schn\"orr (2001) and Lef\`evre and Baillet (2008)
for static image domains. In this work we show the existence and wellposedness
of the corresponding optical flow problem and derive necessary and sufficient
optimality conditions. We demonstrate the functionality of our approach in a
series of experiments using both synthetic and real data.Comment: 26 pages, 6 figure
Learning gradients on manifolds
A common belief in high-dimensional data analysis is that data are
concentrated on a low-dimensional manifold. This motivates simultaneous
dimension reduction and regression on manifolds. We provide an algorithm for
learning gradients on manifolds for dimension reduction for high-dimensional
data with few observations. We obtain generalization error bounds for the
gradient estimates and show that the convergence rate depends on the intrinsic
dimension of the manifold and not on the dimension of the ambient space. We
illustrate the efficacy of this approach empirically on simulated and real data
and compare the method to other dimension reduction procedures.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ206 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
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