Some theory for ordinal embedding

Motivated by recent work on ordinal embedding (In Proceedings of the 27th Conference on Learning Theory (2014) 40–67), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons. We also consider a variant of this problem where only local comparisons are provided. Finally, inspired by (In Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on (2011) 1077–1084 IEEE), we bound the number of such comparisons needed to achieve consistency.