# Convergence rates for ordinal embedding

## TL;DR

This paper establishes optimal convergence rates for 1-dimensional ordinal embedding, linking it to additive number theory, and explores potential rates in higher dimensions through computational experiments.

## Contribution

It provides the first optimal bounds for 1D ordinal embedding convergence rates and connects these bounds to additive number theory, with preliminary insights into higher dimensions.

## Key findings

- Optimal convergence bounds for 1D ordinal embedding
- Examples based on sets with no three-term arithmetic progressions
- Initial computational insights into higher-dimensional convergence rates

## Abstract

We prove optimal bounds for the convergence rate of ordinal embedding (also known as non-metric multidimensional scaling) in the 1-dimensional case. The examples witnessing optimality of our bounds arise from a result in additive number theory on sets of integers with no three-term arithmetic progressions. We also carry out some computational experiments aimed at developing a sense of what the convergence rate for ordinal embedding might look like in higher dimensions.