1,885 research outputs found
Gromov-Monge quasi-metrics and distance distributions
Applications in data science, shape analysis and object classification
frequently require maps between metric spaces which preserve geometry as
faithfully as possible. In this paper, we combine the Monge formulation of
optimal transport with the Gromov-Hausdorff distance construction to define a
measure of the minimum amount of geometric distortion required to map one
metric measure space onto another. We show that the resulting quantity, called
Gromov-Monge distance, defines an extended quasi-metric on the space of
isomorphism classes of metric measure spaces and that it can be promoted to a
true metric on certain subclasses of mm-spaces. We also give precise
comparisons between Gromov-Monge distance and several other metrics which have
appeared previously, such as the Gromov-Wasserstein metric and the continuous
Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive
polynomial-time computable lower bounds for Gromov-Monge distance. These lower
bounds are expressed in terms of distance distributions, which are classical
invariants of metric measure spaces summarizing the volume growth of metric
balls. In the second half of the paper, which may be of independent interest,
we study the discriminative power of these lower bounds for simple subclasses
of metric measure spaces. We first consider the case of planar curves, where we
give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver.
Our results on plane curves are then generalized to higher dimensional
manifolds, where we prove some sphere characterization theorems for the
distance distribution invariant. Finally, we consider several inverse problems
on recovering a metric graph from a collection of localized versions of
distance distributions. Results are derived by establishing connections with
concepts from the fields of computational geometry and topological data
analysis.Comment: Version 2: Added many new results and improved expositio
Locally rich compact sets
We construct a compact metric space that has any other compact metric space
as a tangent, with respect to the Gromov-Hausdorff distance, at all points.
Furthermore, we give examples of compact sets in the Euclidean unit cube, that
have almost any other compact set of the cube as a tangent at all points or
just in a dense sub-set. Here the "almost all compact sets" means that the
tangent collection contains a contracted image of any compact set of the cube
and that the contraction ratios are uniformly bounded. In the Euclidean space,
the distance of sub-sets is measured by the Hausdorff distance. Also the
geometric properties and dimensions of such spaces and sets are studied.Comment: 29 pages, 3 figures. Final versio
Recovering metric from full ordinal information
Given a geodesic space (E, d), we show that full ordinal knowledge on the
metric d-i.e. knowledge of the function D d : (w, x, y, z) 1
d(w,x)d(y,z) , determines uniquely-up to a constant factor-the metric d.
For a subspace En of n points of E, converging in Hausdorff distance to E, we
construct a metric dn on En, based only on the knowledge of D d on En and
establish a sharp upper bound of the Gromov-Hausdorff distance between (En, dn)
and (E, d)
Curvature of sub-Riemannian spaces
To any metric spaces there is an associated metric profile. The
rectifiability of the metric profile gives a good notion of curvature of a
sub-Riemannian space. We shall say that a curvature class is the rectifiability
class of the metric profile. We classify then the curvatures by looking to
homogeneous metric spaces. The classification problem is solved for contact 3
manifolds, where we rediscover a 3 dimensional family of homogeneous contact
manifolds, with a distinguished 2 dimensional family of contact manifolds which
don't have a natural group structure
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