Transportation cost spaces and stochastic trees

TL;DR

This work studies transportation cost spaces $\mathcal{F}(M)$ (Lipschitz free spaces) over finite metric spaces and introduces the stochastic $\ell_1^N$-distortion sd$_1(M)$ as a robust measure of proximity to $\ell_1^N$. By developing a metric characterization that connects sd$_1(M)$ to expected tree distortions under stochastic retractions, the authors provide a framework to approximate optimal transports via stochastic bases. They prove a partial solution linking sd$_1(M)$ to a geodesic-tree structure, and they apply the method to obtain sharp upper bounds for the $\ell_1^N$-distortion of Laakso graphs and of finite hyperbolic approximations of doubling spaces, with explicit bounds such as $sd_1(\mathcal{L}_k) \le 8k$ and $d_1(M_k) \le sd_1(M_k) \le 1+2D\lambda^{\log_2 D}$. Overall, the work provides both a geometric-analytic criterion for $\mathcal{F}(M)$ to resemble $\ell_1^N$ and practical consequences for embedding finite metric spaces into $\ell_1$, with implications for algorithmic transport and metric-geometry constructions.

Abstract

We study transportation cost spaces over finite metric spaces, also known as Lipschitz free spaces. Our work is motivated by a core problem posed by S. Dilworth, D. Kutzarova and M. Ostrovskii, namely, find a condition on a metric space $M$ equivalent to the Banach-Mazur distance between the transportation cost space over $M$ and $\ell_1^N$ of the corresponding dimension, which we call the $\ell_1^N$-distortion of $M$. In this regard, some examples have been studied like the $N\times N$ grid by Naor and Schechtman (2007) and the Laakso and diamond graphs by Dilworth, Kutzarova and Ostrovskii (2020), later studied by Baudier, Gartland and Schlumprecht (2023). We present here three main results. Firstly, we give a partial solution to this problem relating to the tree-like structure of the metric space. For that purpose, we develop a new technique that could potentially lead to a complete solution of the problem and utilize it to find an asymptotically tight upper bound of the $\ell_1^N$-distortion of the Laakso graphs, fully solving an open problem raised by Dilworth, Kutzarova and Ostrovskii. Finally, we apply our technique to prove that finite hyperbolic approximations of doubling metric spaces have uniformly bounded $\ell_1^N$-distortion.

Figures (13)

• Figure 1: On the left side, two optimal transports are shown for the transportation problems $x$ and $y$ separately (supply in black dots and demand in white dots). On the right side, the optimal transport of $x+y$ is shown.
• Figure 2: $F$ is an ordering of a metric space $M$ with 6 elements. Note that the tree $T_1$ is compatible with $F$ but the tree $T_2$ is not since $5\in [0,4)_{T_2}$ and $3\in [0,2)_{T_2}$.
• Figure 3: The meeting point of a path $[x,y]_T$ for a tree $T$.
• Figure 4: Representation of a vector $b_n=\delta_n-\sum_{i<n}\lambda_{n,i}\delta_i$ as a transportation problem.
• Figure 5: The graph $G$ and the basis $(b_n)$ of Example \ref{['examp1']}

Theorems & Definitions (45)

• Theorem 1
• Corollary 1
• Proposition 3.1
• Theorem 3.2
• proof : Proof of Proposition \ref{['propbasisvect']}
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5