$L_1$-distortion of Earth Mover Distances and Transportation Cost Spaces on High Dimensional Grids

TL;DR

The paper establishes a universal lower bound on the distortion of embedding high-dimensional grid-based transportation cost spaces into L1, matching known upper bounds and using a novel Sobolev inequality approach.

Contribution

It introduces a new Sobolev inequality for functions on grids to prove tight bounds on L1-distortion of earth mover and transportation cost spaces.

Findings

Lower bound of Ω(log N) for L1 embedding distortion

Matching upper bound of O(log N) for N-point metric spaces

Novel Sobolev inequality based on dyadic cubes

Abstract

We prove that the distortion of any embedding into $L_1$ of the transportation cost space or earth mover distance over a $d$-dimensional grid $\{1,\dots m\}^d$ is $\Omega(\log N)$, where $N$ is the number of vertices and the implicit constant is universal (in particular, independent of dimension). This lower bound matches the universal upper bound $O(\log N)$ holding for any $N$-point metric space. Our proof relies on a new Sobolev inequality for real-valued functions on the grid, based on random measures supported on dyadic cubes.