abstract: We consider how a closed Riemannian manifold M and its metric tensor g can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse problem of determining (M, g) from limited knowledge on the heat kernel. In the part 1 of the paper, we considered the approximate construction of a smooth manifold in the case when one is given the noisy distances d (x, y) =d (x, y) +ₗ, ₘ for all points x, y X, where X is a -dense subset of M and |ₗ, ₘ|<. In this part 2 of the paper, we consider a similar problem with partial data, that is, the approximate construction of the manifold (M, g) when we are given d (x, y) for x X and y U X, where U is an open subset of M. In addition, we consider the inverse problem of determining the manifold (M, g) with non-negative Ricci curvature from noisy observations of the heat kernel G (y, z, t). We show that a manifold approximating (M, g) can be determined in a stable way, when for some unknown source points zⱼ in X U, we are given the values of the heat kernel G (y, zₖ, t) for y X U and t (0, 1) with a multiplicative noise. We also give a uniqueness result for the inverse problem in the case when the data does not contain noise and consider applications in manifold learning. A novel feature of the inverse problem for the heat kernel is that the set M U containing the sources and the observation set U are disjoint.
Fefferman et al. (Wed,) studied this question.