Kantorovich-Rubinstein Norm and Its Application in the Theory of Lipschitz Spaces

Hanin, Leonid

doi:10.2307/2159251

Cited by 30 publications

(47 citation statements)

References 5 publications

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“…When chosing p = 1 in (7), one obtains another partial optimal transport problem, with | · | as a ground metric. This distance was already introduced by [Han92] in view of the study of Lipschitz spaces. Without providing a proof, it is quite clear that this distance is also obtained by minimizing…”

Section: Previous Work and Connectionsmentioning

confidence: 99%

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

et al. 2016

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This paper defines a new transport metric over the space of non-negative measures. This metric interpolates between the quadratic Wasserstein and the Fisher-Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher-Rao metric, and is averaged with the transportation term. This gives rise to a convex variational problem defining our metric. Our first contribution is a proof of the existence of geodesics (i.e. solutions to this variational problem). We then show that (generalized) optimal transport and Fisher-Rao metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Diracs are made of translating mixtures of Diracs. Lastly, we propose a numerical scheme making use of first order proximal splitting methods and we show an application of this new distance to image interpolation.

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Section: Previous Work and Connectionsmentioning

confidence: 99%

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

et al. 2016

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“…A reasonably rich literature exists on this subject; see [1], [2], [8], [19], [24], [26], [27], [32], [33], [35], [37], [38], [39], [40], [41], [42], [43], [50], [52], [53], [54], [60], [66], [67], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78].…”

Section: Lipschitz Algebrasmentioning

confidence: 99%

“…lip 0 (X) is mainly of interest when it separates points uniformly ( [33], [74]); by ([74], Theorem 3.4) every lip 0 (X) is isometrically isomorphic to some lip 0 (Y) which has this property. Under this hypothesis the double dual of lip 0 (X) is isometrically isomorphic to Lip 0 (X) ( [39], Theorem 4.7; [2], Theorem 3.5).…”

Section: Little Lipschitz Spacesmentioning

confidence: 99%

Lipschitz Algebras and Derivations of von Neumann Algebras

Weaver

1996

Journal of Functional Analysis

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Let M be an abelian von Neumann algebra and E an abelian operator M-bimodule. Then the domain of any ultraweakly closed derivation from M into E is a Lipschitz algebra. Conversely, every Lipschitz algebra can be realized as such a domain. We discuss the relevance of this result to the theory of noncommutative metric spaces.1996 Academic Press, Inc. This paper is the result of an attempt to understand the role of Lipschitz algebras in Alain Connes' theory of noncommutative metric spaces (see [22], Chapter VI). We found that the class of Lipschitz algebras is identical to the class of domains of unbounded derivations of abelian von Neumann algebras (Theorems 9 and 16). The exact statement of this result requires the notion of measurable metric space (see Section I) and a definition of unbounded derivations which is appropriate to the von Neumann algebra context (see Section II).The relation to noncommutative metric theory arises in two ways. First, Lipschitz algebras are dual to metric spaces in much the same way that abelian C*-algebras are dual to topological spaces ([77]; see Section I below), thus creating an analogy with noncommutative topology. Second, unbounded derivations also arise in Connes' theory, although our approach is slightly different (see Section V). In Section V we discuss the implications of our results for the theory of noncommutative metric spaces.We also prove some miscellaneous other results about domains of derivations of von Neumann algebras, in Section II. Although there is a large literature on unbounded derivations of C*-algebras (e.g. see the monographs [9]

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“…In this sense, the extension operator E preserves the modulus of continuity. Theorems 1 and 2 imply an assertion stronger than Proposition 3 in [6], which is used in studying the spaces of Lipschitz functions on (X, d).…”

Section: T--d(~z)mentioning

confidence: 96%

Extension of functions preserving the modulus of continuity

Milman

1997

Math Notes

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ABSTRACT. The problem of the extension of a real-valued function from a subset of a metric space to the entire space is treated. An extension operator preserving the modulus of continuity of a function is proposed and its properties are studied. An application to the problem of the trace of a locally Lipschitz function on a compact subset of a metric space is given.KEY WORDS: extension of a function, modulus of continuity, trace of a locally Lipschitz function.The problem of the extension of a real-valued function from a subset of a metric space to the entire space, preserving the modulus of continuity, is considered. Such problems arise in calculus and geometry (see [1, p. 220 The first class consists of functions with Lipschitz constant not exceeding m, and the second class consists of functions whose modulus of continuity has a prescribed majorant w. The function w need not be a modulus of continuity.The relative Lipschitz constant of a function h: X ~ R at a point x is defined by I } k(h,x) = sup ~:z) zeX, z#x

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Kantorovich-Rubinstein Norm and Its Application in the Theory of Lipschitz Spaces

Cited by 30 publications

References 5 publications

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

Lipschitz Algebras and Derivations of von Neumann Algebras

Extension of functions preserving the modulus of continuity

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