Infinite multidimensional scaling for metric measure spaces*

For a given metric measure space (X, d, mu) we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithm with this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure mu. This limit can be viewed as infinite MDS embedding of the original space, now not anymore into a finite-dimensional space but rather into an infinitedimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show that in many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances).

Automated summary

For a given metric measure space, we can use the multidimensional scaling algorithm to reconstruct finite samples of points into another space. As the number of points in the samples approaches infinity and the density of points approaches a measure, this procedure has a natural limit where the original space is embedded into an infinite-dimensional Hilbert space. However, contrary to common belief, this embedding does not preserve distances in many cases.