Recovering metric from full ordinal information

Thibaut Le Gouic (1, 2)

1 I2M - Institut de Mathématiques de Marseille
2 CS-HSE - Faculty of Computer Science [Moscow]

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).

Domaines: Machine Learning [stat.ML]; Statistiques [math.ST]

Thibaut Le Gouic. Recovering metric from full ordinal information. 2018. ⟨hal-01162490v4⟩