ترغب بنشر مسار تعليمي؟ اضغط هنا

Stochastic order on metric spaces and the ordered Kantorovich monad

74   0   0.0 ( 0 )
 نشر من قبل Paolo Perrone
 تاريخ النشر 2018
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances. We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone affine maps. Considering the category of L-ordered metric spaces as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively. In the unordered case, we had identified the Wasserstein space as the colimit of the spaces of empirical distributions of finite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.



قيم البحث

اقرأ أيضاً

We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. Thi s monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limits of finite samples. The proof relies on a criterion for when an ordinary left Kan extension of lax monoidal functors is a monoidal Kan extension. The colimit characterization allows the development of integration theory and the treatment of measures on spaces of measures, without measure theory. We also show that the category of algebras of the Kantorovich monad is equivalent to the category of closed convex subsets of Banach spaces with short affine maps as morphisms.
Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order co mplexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of interval posets. We give several applications. First, we show the vanishing of higher magnitude homology groups for convex subsets of the Euclidean space. Second, magnitude homology groups carry the information about the diameter of a hole. Third, we construct a finite graph whose $3$rd magnitude homology group has torsion.
157 - Sam van Gool 2010
We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.
An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d rightarrow infty$ is Lipschitz and $left{x_1, dots, x_N right} subset [0,1]^d$, then $$ left| int_{[0,1]^d} f(x) dx - frac{1}{N} sum_{k=1}^{N}{f(x_k)} right| leq le ft| abla f right|_{L^{infty}} cdot W_1left( frac{1}{N} sum_{k=1}^{N}{delta_{x_k}} , dxright),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Movers) Distance. We prove another such inequality with a smaller norm on $ abla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e. $W_{infty} sim N^{-1/d}$. This prompts the question whether these two inequalities are specific instances of an entire underlying family of estimates capturing a duality between transport distance and function space.
148 - Robert Kenny 2015
We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that concerns c overing dimension extends to higher dimensions and to closed shrinkings of finite open covers. To deal with zero-dimensional subspaces uniformly, four operations (relative to the space and a class of subspaces) are defined; these correspond to definitions of inductive and covering dimensions and a countable basis condition. Finally, an effective retract characterisation of zero-dimensionality is proven under an effective compactness condition. In one direction this uses a version of the construction of bilocated sets.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا