A Geometrically Constrained Manifold Embedding for an Extrinsic Gaussian Process

Abstract

We introduce a new framework of local and adaptive manifold embedding for Gaussian regression. The proposed method, which can be generalized on any bounded domain in $\mathbb {R}^n$, is used to construct a smooth vector field from line integral on curves. We prove that optimizing the local shapes from data set leads to a good representation of the generator of a continuous Markov process, which converges in the limit of large data. We explicitly show that the properties of the operator with respect to a geometry are influenced by the constraints and the properties of the covariance function. In this way, we make use of Markov fields to solve a registration problem and place them in a geometric framework. Finally, this locally adaptive embedding can be used with the help of the linear operator to construct conformal mappings or even global diffeomorphisms.