Towards an Algorithmic Realization of Nash’s Embedding Theorem

It is well known from differential geometry that an n-dimensional Riemannian manifold can be isometrically embedded in a Euclidean space of dimension 2n+1 [Nas54]. Though the proof by Nash is intuitive, it is not clear whether such a construction is achievable by an algorithm that only has access to a finite-size sample from the manifold. In this paper, we study Nash’s construction and develop two algorithms for embedding a fairly general class of n-dimensional Riemannian manifolds (initially residing in R) into R (where k only depends on some key manifold properties, such as its intrinsic dimension, its volume, and its curvature) that approximately preserves geodesic distances between all pairs of points. The first algorithm we propose is computationally fast and embeds the given manifold approximately isometrically into about O(2) dimensions (where c is an absolute constant). The second algorithm, although computationally more involved, attempts to minimize the dimension of the target space and (approximately isometrically) embeds the manifold in about O(n) dimensions.