Lower Estimates for $L_1$-Distortion of Transportation Cost Spaces

TL;DR

This work resolves a central question in metric embeddings of transportation-cost spaces by proving sharp $L_1$-distortion bounds. It shows $c_1({\rm TC}({0,...,n}^2)) = \Theta(\log n)$ for planar grids, matching the universal upper bound, and extends to broad recursive graph families via edge-replacement and slash-product constructions, yielding $c_1({\rm TC}(G^{\oslash n})) \ge C^{-1}\log |V(G^{\oslash n})|$. The authors introduce a Sobolev-type inequality on graphs, derive a novel lower-bound machinery based on a pair of conditions $(C1)$–$(C2)$, and reduce the analysis to linear maps through Bourgain discretization. Collectively, these results provide sharp, broadly applicable lower bounds for $EMD$-based distortions and illuminate the geometry of transportation-cost spaces on complex graph families, with concrete constants for diamond and Laakso graphs. The techniques have potential implications for understanding $EMD$-based metrics in computer vision and related discrete-geometry problems.

Abstract

Quantifying the degree of dissimilarity between two probability distributions on a finite metric space is a fundamental task in Computer Science and Computer Vision. A natural dissimilarity measure based on optimal transport is the Earth Mover's Distance (EMD). A key technique for analyzing this metric, pioneered by Charikar (2002) and Indyk and Thaper (2003), involves constructing low-distortion embeddings of EMD(X) into the Lebesgue space $L_1$. It became a key problem to investigate whether the upper bound of $O(\log n)$ can be improved for important classes of metric spaces known to admit low-distortion embeddings into $L_1$. In the context of Computer Vision, grid graphs, especially planar grids, are among the most fundamental. Indyk posed the related problem of estimating the $L_1$-distortion of the space of uniform distributions on $n$-point subsets of $R^2$. The Progress Report, last updated in August 2011, highlighted two key results: first, the work of Khot and Naor (2006) on Hamming cubes, which showed that the $L_1$-distortion for Hamming cubes meets the described above upper estimate, and second, the result of Naor and Schechtman (2007) for planar grids, which established that the $L_1$-distortion of for a planar $n$ by $n$ grid is $Ω(\sqrt{\log n})$. Our first result is the improvement of the lower bound on the $L_1$-distortion for grids to $Ω(\log n)$, matching the universal upper bound up to multiplicative constants. The key ingredient allowing us to obtain these sharp estimates is a new Sobolev-type inequality for scalar-valued functions on the grid graphs. Our method is also applicable to many recursive families of graphs, such as diamond and Laakso graphs. We obtain the sharp distortion estimates of $\log n$ in these cases as well.

Figures (3)

• Figure 1: The first two diamond graphs
• Figure 2: The first Laakso graph $La_1$.
• Figure 3: Left: The decomposition of $Gr_4$ into dyadic subgrids $\{Q_t\}_{t\in T_1}$ indexed by $T_1 = \{1,2,3,4\}$. Center: The decomposition of $Gr_4$ into dyadic subgrids $\{Q_t\}_{t\in T_2}$ indexed by $T_2 = \{1,2,3,4\}^2$. Right: The positive (black) and negative (red) supports of the measures $\mu_t$ for $|t| \leq 2$.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2: Ost05
• Theorem 2.3: Reduction to Simply Connected Sets
• proof
• Theorem 3.1
• proof
• Lemma 3.2