[Paper Review] Realizations of Gromov-Hausdorff Distance

Alexander Ivanov, S.D. Iliadis+1 more|arXiv (Cornell University)|Mar 29, 2016

Medical and Biological Sciences4 references16 citations

TL;DR

This paper establishes the existence of optimal correspondences between any two compact metric spaces, proving that the Gromov–Hausdorff distance is always attained at such correspondences. Using these optimal correspondences, the authors construct explicit isometric embeddings into a common metric space where the Hausdorff distance between the images equals the Gromov–Hausdorff distance, and further show that these correspondences generate shortest curves in the Gromov–Hausdorff space, confirming its geodesic structure via elementary compactness arguments.

ABSTRACT

It is shown that for any two compact metric spaces there exists an "optimal" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric space such that the Gromov-Hausdorff distance between the initial spaces is equal to the Hausdorff distance between their images. Also, the optimal correspondences could be used for constructing the shortest curves in the Gromov-Hausdorff space in exactly the same way as it was done by Alexander Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in arXiv:1504.03830, where it is proved that the Gromov-Hausdorff space is geodesic. Notice that all proofs in the present paper are elementary and use no more than the idea of compactness.

Motivation & Objective

• To prove that the Gromov–Hausdorff distance between any two compact metric spaces is attained at an optimal correspondence.
• To construct explicit realizations of compact metric spaces in a common metric space where the Hausdorff distance between images equals the Gromov–Hausdorff distance.
• To demonstrate that optimal correspondences can be used to construct shortest curves in the Gromov–Hausdorff space, confirming its geodesic property.
• To provide an elementary, compactness-based proof that avoids advanced tools, making the results accessible and self-contained.

Proposed method

• Define a correspondence between compact metric spaces X and Y as a relation where both projections are surjective.
• Introduce the distortion of a correspondence, defined as the supremum of the absolute difference in distances between pairs of related points.
• Prove that the distortion function on the space of compact correspondences is continuous and attains its minimum due to compactness of the space of compact correspondences.
• Show that the Gromov–Hausdorff distance equals half the distortion of an optimal correspondence.
• Construct a pseudometric on the disjoint union X ⊔ Y using an optimal correspondence, leading to a quotient space where the images of X and Y realize the Gromov–Hausdorff distance as a Hausdorff distance.
• Use the optimal correspondence to define a one-parameter family of metric spaces R_t with a metric ρ_t that interpolates linearly between the metrics on X and Y, forming a shortest curve in the Gromov–Hausdorff space.

Experimental results

Research questions

• RQ1 Does the Gromov–Hausdorff distance between any two compact metric spaces always attain its infimum at some correspondence?
• RQ2 Can every pair of compact metric spaces be isometrically embedded into a common metric space such that the Hausdorff distance between their images equals the Gromov–Hausdorff distance?
• RQ3 Can optimal correspondences be used to explicitly construct shortest curves in the Gromov–Hausdorff space?
• RQ4 Is the Gromov–Hausdorff space geodesic, and can this be proven using elementary compactness arguments?

Key findings

• For any two compact metric spaces X and Y, there exists at least one optimal correspondence R ∈ R_opt(X,Y) such that d_GH(X,Y) = (1/2) · dis(R).
• There exists a pseudometric ρ on X ⊔ Y such that d_H(X,Y,ρ) = d_GH(X,Y), providing a direct realization of the Gromov–Hausdorff distance as a Hausdorff distance in a common space.
• There exists a metric space Z and isometric embeddings of X and Y into Z such that the Hausdorff distance between their images equals d_GH(X,Y).
• For any optimal correspondence R, the family of spaces (R, ρ_t) with ρ_t((x,y),(x',y')) = (1−t)|xx'| + t|yy'| forms a shortest curve in the Gromov–Hausdorff space connecting X and Y.
• The proof relies solely on the compactness of the space of compact correspondences and continuity of the distortion function, avoiding advanced analytical tools.
• The results confirm that the Gromov–Hausdorff space is geodesic, and the construction of shortest curves is explicit and constructive via optimal correspondences.