Results 1  10
of
38
On the differentiation of Lipschitz maps from metric measure spaces to Banach spaces
, 2006
"... Abstract. We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting biLipschitz nonembedding theorems for finite dimensional ve ..."
Abstract

Cited by 33 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting biLipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximation. This class of targets includes separable dual spaces. We also observe that there is a straightforward extension of Pansu’s differentiation theory for Lipschitz maps between Carnot groups, [Pan89], to the most general possible class of Banach space targets, those with the RadonNikodym property. Contents
NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
Abstract

Cited by 25 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
A dual characterization of length spaces with application to Dirichlet metric spaces
, 2009
"... ..."
(Show Context)
Bouligand Derivatives and Robustness of Support Vector Machines for Regression
 Journal of Machine Learning Research
, 2008
"... We investigate robustness properties for a broad class of support vector machines with nonsmooth loss functions. These kernel methods are inspired by convex risk minimization in infinite dimensional Hilbert spaces. Leading examples are the support vector machine based on the εinsensitive loss func ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
(Show Context)
We investigate robustness properties for a broad class of support vector machines with nonsmooth loss functions. These kernel methods are inspired by convex risk minimization in infinite dimensional Hilbert spaces. Leading examples are the support vector machine based on the εinsensitive loss function, and kernel based quantile regression based on the pinball loss function. Firstly, we propose with the Bouligand influence function (BIF) a modification of F.R. Hampel’s influence function. The BIF has the advantage of being positive homogeneous which is in general not true for Hampel’s influence function. Secondly, we show that many support vector machines based on a Lipschitz continuous loss function and a bounded kernel have a bounded BIF and are thus robust in the sense of robust statistics based on influence functions.
Schied: Rademacher’s theorem on configuration spaces and applications
, 1998
"... Abstract: We consider an L 2Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρLipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Abstract: We consider an L 2Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρLipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular fulfilled by the classical Poisson measures and by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of ρ for a set to be exceptional. This result immediately implies, for instance, a quasisure version of the spatial ergodic theorem. We also show that ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form. 0. Introduction. Let ΓX be the configuration space over a Riemannian manifold X. In this paper, we consider a class of probability measures on ΓX, which in particular contains certain Ruelle type Gibbs measures and mixed Poisson measures. Using a natural ‘nonflat ’ geometric structure of ΓX, recently analyzed in Albeverio, Kondratiev and Röckner (1996a),
Geometric and transformational properties of Lipschitz domains, SemmesKenigToro domains, and other classes of finite perimeter domains
 Journal of Geometric Analysis
, 2007
"... In the first part of this paper we give intrinsic characterizations of the classes of Lipschitz and C 1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversa ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
(Show Context)
In the first part of this paper we give intrinsic characterizations of the classes of Lipschitz and C 1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C 1. In the second part of the paper, we study the invariance of various classes of domains of locally finite perimeter under biLipschitz and C 1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chordarc domains with vanishing constant, in the literature) is stable under C 1 diffeomorphisms. A number of other applications are also presented. 1
A MorseSard theorem for the distance function on Riemannian manifolds
 Manuscr. Math
"... Abstract. We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka. 1. ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka. 1.
Differential Geometric Aspects of Alexandrov Spaces
, 1997
"... We summarize the results on the differential geometric structure of Alexandrov spaces developed in [Otsu and Shioya 1994; Otsu 1995; Otsu and Tanoue a]. We discuss Riemannian and second differentiable structure and Jacobi fields on Alexandrov spaces of curvature bounded below or above. 1 ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
We summarize the results on the differential geometric structure of Alexandrov spaces developed in [Otsu and Shioya 1994; Otsu 1995; Otsu and Tanoue a]. We discuss Riemannian and second differentiable structure and Jacobi fields on Alexandrov spaces of curvature bounded below or above. 1
Metric and w ∗ differentiability of pointwise Lipschitz mappings, submitted (available electronically at http://www.karlin.mff.cuni.cz/kmapreprints
"... Abstract. We prove that for every function f: X → Y, where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈ Ã such that f is Gâteaux differentiable at all x ∈ S(f) \ A, where S(f) is the set of points where f is pointwiseLipschitz. This improves a result of Bo ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Abstract. We prove that for every function f: X → Y, where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈ Ã such that f is Gâteaux differentiable at all x ∈ S(f) \ A, where S(f) is the set of points where f is pointwiseLipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every Kmonotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to ˜ C; this improves a result due to Borwein and Wang. Another corollary is that if X is Asplund, f: X → R cone monotone, g: X → R continuous convex, then there exists a point in X, where f is Hadamard differentiable and g is Fréchet differentiable. 1.