  We consider the problem of reconstructing an embedding of a compact connected
Riemannian manifold in a Euclidean space up to an almost isometry, given the
information on intrinsic distances between points from its ``sufficiently
large'' subset. This is one of the classical manifold learning problems. It
happens that the most popular methods to deal with such a problem, with a long
history in data science, namely, the classical Multidimensional scaling (MDS)
and the Maximum variance unfolding (MVU), actually miss the point and may
provide results very far from an isometry; moreover, they may even give no
bi-Lipshitz embedding. We will provide an easy variational formulation of this
problem, which leads to an algorithm always providing an almost isometric
embedding with the distortion of original distances as small as desired (the
parameter regulating the upper bound for the desired distortion is an input
parameter of this algorithm).
