805 research outputs found
Computing the Similarity Between Moving Curves
In this paper we study similarity measures for moving curves which can, for
example, model changing coastlines or retreating glacier termini. Points on a
moving curve have two parameters, namely the position along the curve as well
as time. We therefore focus on similarity measures for surfaces, specifically
the Fr\'echet distance between surfaces. While the Fr\'echet distance between
surfaces is not even known to be computable, we show for variants arising in
the context of moving curves that they are polynomial-time solvable or
NP-complete depending on the restrictions imposed on how the moving curves are
matched. We achieve the polynomial-time solutions by a novel approach for
computing a surface in the so-called free-space diagram based on max-flow
min-cut duality
Quantum Hellinger distances revisited
This short note aims to study quantum Hellinger distances investigated
recently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a
particular emphasis on barycenters. We introduce the family of generalized
quantum Hellinger divergences, that are of the form where is an arbitrary
Kubo-Ando mean, and is the weight of We note that these
divergences belong to the family of maximal quantum -divergences, and hence
are jointly convex and satisfy the data processing inequality (DPI). We derive
a characterization of the barycenter of finitely many positive definite
operators for these generalized quantum Hellinger divergences. We note that the
characterization of the barycenter as the weighted multivariate -power
mean, that was claimed in the work of Bhatia et al. mentioned above, is true in
the case of commuting operators, but it is not correct in the general case.Comment: v2: Section 4 on the commutative case, and Subsection 5.2 on a
possible measure of non-commutativity added, as well as references to the
maximal quantum -divergence literature; v3: Section 4 on the commutative
case improved, and the proposed measure of non-commutativiy changed
accordingly; v4: accepted manuscript versio
Fast Frechet Distance Between Curves With Long Edges
Computing the Fr\'echet distance between two polygonal curves takes roughly
quadratic time. In this paper, we show that for a special class of curves the
Fr\'echet distance computations become easier. Let and be two polygonal
curves in with and vertices, respectively. We prove four
results for the case when all edges of both curves are long compared to the
Fr\'echet distance between them: (1) a linear-time algorithm for deciding the
Fr\'echet distance between two curves, (2) an algorithm that computes the
Fr\'echet distance in time, (3) a linear-time
-approximation algorithm, and (4) a data structure that supports
-time decision queries, where is the number of vertices of
the query curve and the number of vertices of the preprocessed curve
Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity
We propose a new concept of generalized differentiation of set-valued maps
that captures the first order information. This concept encompasses the
standard notions of Frechet differentiability, strict differentiability,
calmness and Lipschitz continuity in single-valued maps, and the Aubin property
and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen
the relationship between the Aubin property and coderivatives, and study how
metric regularity and open covering can be refined to have a directional
property similar to our concept of generalized differentiation. Finally, we
discuss the relationship between the robust form of generalization
differentiation and its one sided counterpart.Comment: This submission corrects errors from the previous version after
referees' comments. Changes are in Proposition 2.4, Proposition 4.12, and
Sections 7 and
A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems
In this paper, we prove a Pontryagin Maximum Principle for constrained
optimal control problems in the Wasserstein space of probability measures. The
dynamics, is described by a transport equation with non-local velocities and is
subject to end-point and running state constraints. Building on our previous
work, we combine the classical method of needle-variations from geometric
control theory and the metric differential structure of the Wasserstein spaces
to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl
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