Kernel Regression on Manifold Valued Data

We consider an unknown smooth function which maps high-dimensional inputs to multidimensional outputs and whose domain of definition is an unknown low-dimensional input manifold embedded in an ambient high-dimensional input space. Given a training dataset with "input-output" pairs, Regression with Manifold Valued Inputs problem is to estimate the unknown function and its Jacobian matrix. Previously proposed solutions are very computationally expensive. The paper presents a new geometrically motivated kernel regression method for solving the considered problem with a much lower computational complexity while preserving accuracy.