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13
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 ..."
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Cited by 13 (2 self)
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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 ..."
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Cited by 12 (0 self)
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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.
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 ..."
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Cited by 8 (3 self)
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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),
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 ..."
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Cited by 5 (1 self)
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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.
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 ..."
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Cited by 5 (3 self)
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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.
A dual characterization of length spaces with application to Dirichlet metric spaces
"... We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1Lipschitz functions form a sheaf. ..."
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Cited by 4 (3 self)
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We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1Lipschitz functions form a sheaf.
Abstract. Notes on the classical frequency formula for harmonic functions
"... For r> 0 let Br denote the Euclidean ball with radius r centered at the origin of Rn. For a continuously differentiable function u in B1 and 0 < r < 1, consider the differentiable functions D(r), I(r) and F (r), associated to u, given by ..."
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For r> 0 let Br denote the Euclidean ball with radius r centered at the origin of Rn. For a continuously differentiable function u in B1 and 0 < r < 1, consider the differentiable functions D(r), I(r) and F (r), associated to u, given by
Differential Geometric Aspects of Alexandrov Spaces
"... Abstract. 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. ..."
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Abstract. 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.
ON GÂTEAUX DIFFERENTIABILITY OF POINTWISE LIPSCHITZ MAPPINGS
, 2006
"... 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 ..."
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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.