Some theory for ordinal embedding

TL;DR

This paper provides theoretical guarantees and convergence rates for ordinal embedding methods based on distance comparisons, including local comparisons, and quantifies the number of comparisons needed for consistency.

Contribution

It derives large sample consistency results, convergence rates, and comparison bounds for ordinal embedding, extending previous work to local comparisons.

Findings

Established large sample consistency for ordinal embedding.

Derived convergence rates for embedding accuracy.

Bounded the number of comparisons needed for consistent embedding.

Abstract

Motivated by recent work on ordinal embedding (Kleindessner and von Luxburg, 2014), 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 (Jamieson and Nowak, 2011), we bound the number of such comparisons needed to achieve consistency.