1,794 research outputs found
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
Error Bounds for Piecewise Smooth and Switching Regression
The paper deals with regression problems, in which the nonsmooth target is
assumed to switch between different operating modes. Specifically, piecewise
smooth (PWS) regression considers target functions switching deterministically
via a partition of the input space, while switching regression considers
arbitrary switching laws. The paper derives generalization error bounds in
these two settings by following the approach based on Rademacher complexities.
For PWS regression, our derivation involves a chaining argument and a
decomposition of the covering numbers of PWS classes in terms of the ones of
their component functions and the capacity of the classifier partitioning the
input space. This yields error bounds with a radical dependency on the number
of modes. For switching regression, the decomposition can be performed directly
at the level of the Rademacher complexities, which yields bounds with a linear
dependency on the number of modes. By using once more chaining and a
decomposition at the level of covering numbers, we show how to recover a
radical dependency. Examples of applications are given in particular for PWS
and swichting regression with linear and kernel-based component functions.Comment: This work has been submitted to the IEEE for possible publication.
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Approximations of Semicontinuous Functions with Applications to Stochastic Optimization and Statistical Estimation
Upper semicontinuous (usc) functions arise in the analysis of maximization
problems, distributionally robust optimization, and function identification,
which includes many problems of nonparametric statistics. We establish that
every usc function is the limit of a hypo-converging sequence of piecewise
affine functions of the difference-of-max type and illustrate resulting
algorithmic possibilities in the context of approximate solution of
infinite-dimensional optimization problems. In an effort to quantify the ease
with which classes of usc functions can be approximated by finite collections,
we provide upper and lower bounds on covering numbers for bounded sets of usc
functions under the Attouch-Wets distance. The result is applied in the context
of stochastic optimization problems defined over spaces of usc functions. We
establish confidence regions for optimal solutions based on sample average
approximations and examine the accompanying rates of convergence. Examples from
nonparametric statistics illustrate the results
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