20 research outputs found
Stochastic Development Regression on Non-Linear Manifolds
We introduce a regression model for data on non-linear manifolds. The model
describes the relation between a set of manifold valued observations, such as
shapes of anatomical objects, and Euclidean explanatory variables. The approach
is based on stochastic development of Euclidean diffusion processes to the
manifold. Defining the data distribution as the transition distribution of the
mapped stochastic process, parameters of the model, the non-linear analogue of
design matrix and intercept, are found via maximum likelihood. The model is
intrinsically related to the geometry encoded in the connection of the
manifold. We propose an estimation procedure which applies the Laplace
approximation of the likelihood function. A simulation study of the performance
of the model is performed and the model is applied to a real dataset of Corpus
Callosum shapes
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
Zero-Shot Domain Adaptation via Kernel Regression on the Grassmannian
Most visual recognition methods implicitly assume the data distribution
remains unchanged from training to testing. However, in practice domain shift
often exists, where real-world factors such as lighting and sensor type change
between train and test, and classifiers do not generalise from source to target
domains. It is impractical to train separate models for all possible situations
because collecting and labelling the data is expensive. Domain adaptation
algorithms aim to ameliorate domain shift, allowing a model trained on a source
to perform well on a different target domain. However, even for the setting of
unsupervised domain adaptation, where the target domain is unlabelled,
collecting data for every possible target domain is still costly. In this
paper, we propose a new domain adaptation method that has no need to access
either data or labels of the target domain when it can be described by a
parametrised vector and there exits several related source domains within the
same parametric space. It greatly reduces the burden of data collection and
annotation, and our experiments show some promising results.Comment: Accepted to BMVC 2015 Workshop on Differential Geometry in Computer
Vision (DIFF-CV
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