17 research outputs found
Calibrations for minimal networks in a covering space setting
In this paper we define a notion of calibration for an equivalent approach to
the classical Steiner problem in a covering space setting and we give some
explicit examples. Moreover we introduce the notion of calibration in families:
the idea is to divide the set of competitors in a suitable way, defining an
appropriate (and weaker) notion of calibration. Then, calibrating the candidate
minimizers in each family and comparing their perimeter, it is possible to find
the minimizers of the minimization problem. Thanks to this procedure we prove
the minimality of the Steiner configurations spanning the vertices of a regular
hexagon and of a regular pentagon
On different notions of calibrations for minimal partitions and minimal networks in
Calibrations are a possible tool to validate the minimality of a certain
candidate. They have been introduced in the context of minimal surfaces and
adapted to the case of Steiner problem in several variants. Our goal is to
compare the different notions of calibrations for the Steiner Problem and for
planar minimal partitions. The paper is then complemented with remarks on the
convexification of the problem, on non-existence of calibrations and on
calibrations in families
A General Theory for Exact Sparse Representation Recovery in Convex Optimization
In this paper, we investigate the recovery of the sparse representation of
data in general infinite-dimensional optimization problems regularized by
convex functionals. We show that it is possible to define a suitable
non-degeneracy condition on the minimal-norm dual certificate, extending the
well-established non-degeneracy source condition (NDSC) associated with total
variation regularized problems in the space of measures, as introduced in
(Duval and Peyr\'e, FoCM, 15:1315-1355, 2015). In our general setting, we need
to study how the dual certificate is acting, through the duality product, on
the set of extreme points of the ball of the regularizer, seen as a metric
space. This justifies the name Metric Non-Degenerate Source Condition (MNDSC).
More precisely, we impose a second-order condition on the dual certificate,
evaluated on curves with values in small neighbourhoods of a given collection
of n extreme points. By assuming the validity of the MNDSC, together with the
linear independence of the measurements on these extreme points, we establish
that, for a suitable choice of regularization parameters and noise levels, the
minimizer of the minimization problem is unique and is uniquely represented as
a linear combination of n extreme points. The paper concludes by obtaining
explicit formulations of the MNDSC for three problems of interest. First, we
examine total variation regularized deconvolution problems, showing that the
classical NDSC implies our MNDSC, and recovering a result similar to (Duval and
Peyr\'e, FoCM, 15:1315-1355, 2015). Then, we consider 1-dimensional BV
functions regularized with their BV-seminorm and pairs of measures regularized
with their mutual 1-Wasserstein distance. In each case, we provide explicit
versions of the MNDSC and formulate specific sparse representation recovery
results
Regularization graphs—a unified framework for variational regularization of inverse problems
We introduce and study a mathematical framework for a broad class of
regularization functionals for ill-posed inverse problems: Regularization
Graphs. Regularization graphs allow to construct functionals using as building
blocks linear operators and convex functionals, assembled by means of operators
that can be seen as generalizations of classical infimal convolution operators.
This class of functionals exhaustively covers existing regularization
approaches and it is flexible enough to craft new ones in a simple and
constructive way. We provide well-posedness and convergence results with the
proposed class of functionals in a general setting. Further, we consider a
bilevel optimization approach to learn optimal weights for such regularization
graphs from training data. We demonstrate that this approach is capable of
optimizing the structure and the complexity of a regularization graph,
allowing, for example, to automatically select a combination of regularizers
that is optimal for given training data
Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms
A precise characterization of the extremal points of sublevel sets of
nonsmooth penalties provides both detailed information about minimizers, and
optimality conditions in general classes of minimization problems involving
them. Moreover, it enables the application of accelerated generalized
conditional gradient methods for their efficient solution. In this manuscript,
this program is adapted to the minimization of a smooth convex fidelity term
which is augmented with an unbalanced transport regularization term given in
the form of a generalized Kantorovich-Rubinstein norm for Radon measures. More
precisely, we show that the extremal points associated to the latter are given
by all Dirac delta functionals supported in the spatial domain as well as
certain dipoles, i.e., pairs of Diracs with the same mass but with different
signs. Subsequently, this characterization is used to derive precise
first-order optimality conditions as well as an efficient solution algorithm
for which linear convergence is proved under natural assumptions. This
behaviour is also reflected in numerical examples for a model problem
A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients
We study measure-valued solutions of the inhomogeneous continuity equation
where the coefficients
and are of low regularity. A new superposition principle is proven for
positive measure solutions and coefficients for which the recently-introduced
dynamic Hellinger-Kantorovich energy is finite. This principle gives a
decomposition of the solution into curves
that satisfy the characteristic system ,
in an appropriate sense. In particular, it
provides a generalization of existing superposition principles to the
low-regularity case of where characteristics are not unique with respect to
. Two applications of this principle are presented. First, uniqueness of
minimal total-variation solutions for the inhomogeneous continuity equation is
obtained if characteristics are unique up to their possible vanishing time.
Second, the extremal points of dynamic Hellinger-Kantorovich-type regularizers
are characterized. Such regularizers arise, e.g., in the context of dynamic
inverse problems and dynamic optimal transport