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

Chris Gartland, Mikhail Ostrovskii, Yuval Rabani, Robert Young

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 $Ω(\log N)$, 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.

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

Abstract

We prove that the distortion of any embedding into of the transportation cost space or earth mover distance over a -dimensional grid is , where is the number of vertices and the implicit constant is universal (in particular, independent of dimension). This lower bound matches the universal upper bound holding for any -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.
Paper Structure (11 sections, 13 theorems, 81 equations)

This paper contains 11 sections, 13 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. New and prior results
  3. Notation and background
  4. Organization
  5. Construction of Random Measures
  6. Lower Bound on Expected TC norm
  7. Proving the Sobolev Inequality
  8. A Sobolev inequality for indicator functions
  9. From Indicator Functions to General Functions
  10. Sobolev Inequality Implies L1-Distortion
  11. Acknowledgments

Key Result

Theorem 1.1

If $d \geq 2$, then any embedding of ${\rm TC} (G)$ into $L_1$ has bi-Lipschitz distortion $\Omega(\log N) = \Omega(d\log m)$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1: Sobolev inequality for $\nu_k$
  • Theorem 4.2: Isoperimetric Inequalities
  • ...and 12 more