Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups, with Applications to Earth Mover’s Distance

William Leeb, Ronald Coifman

Research output: Contribution to journalArticle

8 Scopus citations

Abstract

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.

Original languageEnglish (US)
Pages (from-to)910-953
Number of pages44
JournalJournal of Fourier Analysis and Applications
Volume22
Issue number4
DOIs
StatePublished - Aug 1 2016

Keywords

  • Dominating mixed derivatives
  • Earth Mover’s Distance
  • Heat kernel
  • Hölder
  • Lipschitz
  • Semigroup

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