Recovering metric from full ordinal information
This addresses a theoretical problem in metric geometry for researchers, providing foundational insights into metric recovery from limited data.
The paper tackles the problem of reconstructing a metric from only ordinal comparisons of distances, proving that such ordinal information uniquely determines the metric up to a constant factor, and constructs an approximate metric with a sharp bound on the Gromov-Hausdorff distance for finite subspaces.
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).