A semidefinite program for least distortion embeddings of flat tori into Hilbert spaces

TL;DR

This paper introduces an infinite-dimensional semidefinite program to compute minimal distortion embeddings of flat tori into Hilbert spaces, improving bounds and characterizing embeddings for various dimensions.

Contribution

It develops a novel semidefinite programming approach for flat tori embeddings, providing new bounds and explicit solutions for 2D cases.

Findings

Improved lower bounds on minimal distortion of flat tori embeddings

Finite-dimensional embeddings exist for all flat tori

Optimal embeddings identified for 2-dimensional flat tori

Abstract

We derive and analyze an infinite-dimensional semidefinite program which computes least distortion embeddings of flat tori $\mathbb{R}^n/L$, where $L$ is an $n$-dimensional lattice, into Hilbert spaces. This enables us to provide a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an $n$-dimensional flat torus. As further applications we prove that every $n$-dimensional flat torus has a finite dimensional least distortion embedding, that the standard embedding of the standard tours is optimal, and we determine least distortion embeddings of all $2$-dimensional flat tori.