385 research outputs found
Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers
In this work, using Moreau envelopes, we define a complete metric for the set
of proper lower semicontinuous convex functions. Under this metric, the
convergence of each sequence of convex functions is epi-convergence. We show
that the set of strongly convex functions is dense but it is only of the first
category. On the other hand, it is shown that the set of convex functions with
strong minima is of the second category
Metrics and Isometries for Convex Functions
We introduce a class of functional analogs of the symmetric difference metric
on the space of coercive convex functions on with
full-dimensional domain. We show that convergence with respect to these metrics
is equivalent to epi-convergence. Furthermore, we give a full classification of
all isometries with respect to some of the new metrics. Moreover, we introduce
two new functional analogs of the Hausdorff metric on the spaces of coercive
convex functions and super-coercive convex functions, respectively, and prove
equivalence to epi-convergence
Homogenization of a capillary phenomena
We study the height of a liquid in a tube when it contains a great number of thin vertical bars and when its border is finely strained. For this, one uses an epi-convergence method
Convergence of Minima of Integral Functionals, with Applications to Optimal Control and Stochastic Optimization
Epi-convergence of integral functionals is derived under new conditions that can be used in the infinite dimensional case. Applications include: the convergence of the solutions of approximating optimal control problems and of stochastic optimization problems
Scale space consistency of piecewise constant least squares estimators -- another look at the regressogram
We study the asymptotic behavior of piecewise constant least squares
regression estimates, when the number of partitions of the estimate is
penalized. We show that the estimator is consistent in the relevant metric if
the signal is in , the space of c\`{a}dl\`{a}g functions equipped
with the Skorokhod metric or equipped with the supremum metric.
Moreover, we consider the family of estimates under a varying smoothing
parameter, also called scale space. We prove convergence of the empirical scale
space towards its deterministic target.Comment: Published at http://dx.doi.org/10.1214/074921707000000274 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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