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

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.

Two-sample statistics based on anisotropic kernels

“…π(r, r )dµ R (r ) = h p (r), π(r, r )dµ R (r) = h q (r ), where h p , h q are the population histograms, and d R (r, r ) is certain metric on reference set. It may also be possible to construct a metric which is equivalent to the Wasserstein metric by measuring the difference at reference points across multiple scales of covariance matrices, as is done with with Haar wavelet [24] and with diffusion kernels [18]. Efficient estimation scheme of the Wasserstein metric needs to be developed as well as the consistency analysis with n samples.…”

People mover's distance: Class level geometry using fast pairwise data adaptive transportation costs

“…We defer choice of d : X ×X → R + to Section 2.2, and focus here on fast approximations to EMD. We focus on the tree approximation to EMD, which is strongly equivalent to the true EMD for any metric d(x, y) α for α < 1 [8]. Leeb builds this approximation by constructing a partition tree on the feature space, given that it is equipped with a distance metric d :…”

“…Because it is required to calculate every pairwise distance between classes, computation time becomes a major hurdle. To surmount the barrier, we instead focus on an approximate earth mover's distance based on multi scale tree distance, as originally proposed by [8,12], which can be computed in linear time. The use of localized binning to create a distance metric has been used in a variety of contexts, such as in the linguistic bag-of-words models [14] and topic models [13], in which synonyms are grouped together before defining distances between collections of words.…”

A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup

“…For examples of theoretical results involving diffusion semigroups, the interested reader may refer to Sturm [2] and Wu [3]. Some recent applications of diffusion semigroups to dimensionality reduction, data representation, multiscale analysis of complex structures, and the definition and efficient computation of natural diffusion distances can be found in, e.g., [4][5][6][7][8][9][10][11].…”

“…As part of their work in [7,11], Coifman and Leeb introduce a family of multiscale diffusion distances and establish quantitative results about the equivalence of a bounded function being Lipschitz, and the rate of convergence of to , as → 0 + (we are discussing some of their results using a continuous time for convenience; most of Coifman's and Leeb's derivations are done for dyadically discretized times. Moreover, most of the authors' results are in fact established without the assumption of symmetry and under the weaker condition than positivity of the kernel, namely, an appropriate 1 integrability statement (see [11])). To prove the implication that Lipschitz implies an appropriate estimate on the rate of convergence, Coifman and Leeb make a quantitative assumption about the decay of sup ∫ ( , ) ( , ) , as → 0 + ,…”