I am given two metric spaces as two arrays of the same size. Each one is supposed to represent distance between vertices on a mesh in R^3. The meshes are assumed to have the same number of vertices and the correspondence betweeen the vertices is also given. Is there a way to find the a meaningful distance between these two matrices (other than the trivial ones)? I would like the distance be invariant under scaling : in other words if the first mesh's is just a scaling of the other mesh then I want the distance between the two corresponding metric spaces to be zero (even though the distances in the matrices have also been scaled).


I know that Hausdorff distance measures the distance between two sets in the same metric space but note here that my question is a little different so that notion is not exactly useful. 


Thanks.



Distance between two metric spaces
Persistent Homology can be used for this purpose. Given two metric spaces $X$ and $Y$, then one can measure the distance between their persistence diagrams $PD\_i(X)$ and $PD\_i(Y)$ where $i \geq 0$. There are many distances that can be defined on the space of persistence diagrams such as the [bottleneck distance](http://gudhi.gforge.inria.fr/doc/latest/group__bottleneck__distance.html#:%7E:text=The%20bottleneck%20distance%20is%20the,Persistence%20Diagrams%22%20%5B34%5D.) of the [Wasserstein distance](https://gudhi.inria.fr/python/3.1.0.rc1/wasserstein_distance_user.html). The distance between the persistence diagram is much more efficient than the Hausdorff distance because usually the persistence diagram contains less data than the original spaces.


