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- W2897152313 abstract "Applications in shape analysis and object classification often require maps between metric spaces which preserve geometry as much as possible. In this paper, we combine the Monge formulation of optimal transport with the Gromov-Hausdorff 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. Finally, we derive polynomial-time computable lower bounds for Gromov-Monge distance. These lower bounds are expressed in terms of classical invariants of mm-spaces called distance distributions. 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. Next we show that one of our lower bounds distinguishes metric trees locally---trees which lie sufficiently close to one another in Gromov-Hausdorff distance are always distinguished---and generically with respect to a natural measure on the space of trees." @default.
- W2897152313 created "2018-10-26" @default.
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- W2897152313 date "2018-10-23" @default.
- W2897152313 modified "2023-09-27" @default.
- W2897152313 title "Gromov-Monge quasi-metrics and distance distributions" @default.
- W2897152313 hasPublicationYear "2018" @default.
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