Recovering metric from full ordinal information

TL;DR

This paper demonstrates that full ordinal information about a geodesic space's metric uniquely determines the metric up to a constant, and provides a method to approximate the metric on finite subspaces with explicit bounds.

Contribution

It introduces a way to recover the metric from ordinal data and constructs an approximating metric on finite subspaces with proven convergence bounds.

Findings

Full ordinal knowledge determines the metric up to a constant.

Constructed metrics on finite subspaces converge to the true metric.

Established sharp bounds on Gromov-Hausdorff distance for approximations.

Abstract

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) $\rightarrow$ 1 d(w,x)$\le$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).