Coarse and bi-Lipschitz embeddability of subspaces of the Gromov-Hausdorff space into Hilbert spaces

TL;DR

The paper addresses when subspaces of the Gromov-Hausdorff space can be coarsely or bi-Lipschitz embedded into Hilbert spaces. Using coarse geometry and stable invariants, it links embeddability to large-scale dimension and obstruction tools, notably via the distance-set map and Ma j hr–Vitter–Wenk-type results for line-subspaces. It proves that $\mathcal{GH}^{\le n}$ has asymptotic dimension $\frac{n(n-1)}{2}$ and thus coarsely embeds into a Hilbert space, while $\mathcal{GH}^{<\omega}$ cannot coarsely embed into any uniformly convex Banach space, implying non-embeddability into Hilbert spaces; further, even diameter-bounded finite subspaces $\mathcal{EH}_{[0,R]}^{<\omega}$ cannot be bi-Lipschitz embedded into finite-dimensional Hilbert spaces due to infinite Assouad dimension. The results delineate a boundary between coarse and bi-Lipschitz embeddability for GH subspaces and have implications for the use of stable invariants in topological data analysis, clarifying when such invariants can yield effective Euclidean representations at large scales.

Abstract

In this paper, we discuss the embeddability of subspaces of the Gromov-Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension $n(n-1)/2$. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both non-embeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean-Hausdorff distance.

Figures (4)

• Figure 1: A representation of the uniformly bounded cover $\mathcal{V}=\mathcal{V}_0\cup\mathcal{V}_1$ showing that $\mathop{\mathrm{asdim}}\nolimits\mathbb{R}_{\geq 0}\leq 1$. In red, the elements of $\mathcal{V}_0=\{V^0_k\mid k\in\mathbb{N}\}$ and the subsets contained in the family $\mathcal{V}_1=\{V_k^1\mid k\in\mathbb{N}\}$ are in blue.
• Figure 2: A representation of the map $T^{-1}\colon\mathbb{N}\setminus\{0\}\to(\mathbb{N}\setminus\{0\})^2$.
• Figure 3: A representation of the images of two points $(x_i)_i,(y_i)_i\in[0,m]^n$ along $\varphi_m^n$. The subset $\varphi_m^n((x_i)_i)$ is given by the red dots and the black dot, while $\varphi_m^n((y_i)_i)$ consists of the blue dots and the black dot. In the picture, we can see that $d_H(\varphi_m^n((x_i)_i),\varphi_m^n((y_i)_i))=d_m^n((x_i)_i,(y_i)_i)$.
• Figure 4: A representation of the subsets $A$, $A_1$ and $A_i$ defined in the proof of Proposition \ref{['prop:dimA']}. The distinctive points $l-s\in A_1$ and $il-s\in A_i$ are emphasised in red. The light grey strips are meant to visualise the fact that $d_H(A,A_1)<r$ and $d_H(A,A_i)<r$.

Theorems & Definitions (44)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.1
• Definition 2.2
• Definition 2.3: ChoMem3
• Definition 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7