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

2026-02-22

AI Summary

This study addresses the problem of minimizing distortion when embedding the Earth Mover Distance (EMD) over high-dimensional grids into ℓ₁ space. By constructing random measures supported on bipartite cubes and leveraging tools from harmonic analysis, metric embedding theory, and dyadic decomposition, the authors establish a new Sobolev-type inequality tailored to high-dimensional grids. This inequality leads to the first proof that the distortion of any embedding of EMD into ℓ₁ is lower-bounded by Ω(log N), where the implicit constant is independent of the dimension. This lower bound matches the universal upper bound of O(log N) for embeddings of arbitrary N-point metric spaces into ℓ₁, thereby fully characterizing the tight asymptotic bounds for this fundamental embedding problem.

Technology Category

Machine Learning: Learning with ManifoldsConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Learning to Search

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSecurity and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• L1-distortion
• Earth Mover Distance
• Transportation Cost Space
• High Dimensional Grids
• Metric Embedding

Innovation

• Earth Mover Distance
• L1 embedding
• transportation cost space
• Sobolev inequality
• distortion lower bound