  We introduce an estimator for distances in a compact Riemannian manifold
based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper
bound the error in the estimate of manifold distances, or more precisely an
estimate of a spectrally truncated variant of manifold distance of interest in
non-commutative geometry (cf. [Connes and Suijelekom, 2020]), in terms of
spectral errors in the graph Laplacian estimates and, implicitly, several
geometric properties of the manifold. A consequence is a proof of consistency
for (untruncated) manifold distances. The estimator resembles, and in fact its
convergence properties are derived from, a special case of the Kontorovic dual
reformulation of Wasserstein distance known as Connes' Distance Formula.
