21,904 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
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
Regularized estimation of linear functionals of precision matrices for high-dimensional time series
This paper studies a Dantzig-selector type regularized estimator for linear
functionals of high-dimensional linear processes. Explicit rates of convergence
of the proposed estimator are obtained and they cover the broad regime from
i.i.d. samples to long-range dependent time series and from sub-Gaussian
innovations to those with mild polynomial moments. It is shown that the
convergence rates depend on the degree of temporal dependence and the moment
conditions of the underlying linear processes. The Dantzig-selector estimator
is applied to the sparse Markowitz portfolio allocation and the optimal linear
prediction for time series, in which the ratio consistency when compared with
an oracle estimator is established. The effect of dependence and innovation
moment conditions is further illustrated in the simulation study. Finally, the
regularized estimator is applied to classify the cognitive states on a real
fMRI dataset and to portfolio optimization on a financial dataset.Comment: 44 pages, 4 figure
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