RECONSTRUCTION AND INTERPOLATION OF MANIFOLDS II: INVERSE PROBLEMS WITH PARTIAL DATA FOR DISTANCES OBSERVATIONS AND FOR THE HEAT KERNEL

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^e(x, y) = d(x, y) + εx,y for all points x, y ∈ X, where X is a δ-dense subset of M and |εx,y| < δ. 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^e(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,z_k,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.