Statistical Learning on Manifold-Valued Data

Regression on manifolds problem is to estimate an unknown smooth function f that maps p-dimensional manifold-valued inputs, whose values lie on unknown Input manifold M of lower dimensionality q < p embedded in an ambient high-dimensional input space R-p, to m-dimensional outputs from training sample consisting of given 'input-output' pairs. We consider this problem in which Jacobian J(f)(X) of function f and Input manifold M should be also estimated. The paper presents a new geometrically motivated method for estimating a triple (f(X), J(f)(X), M) from given sample. The proposed solution is based on solving a Tangent bundle manifold learning problem for specific unknown Regression manifold embedded in input-output space Rp+m and consisting of input-output pairs (X, f(X)), X is an element of M.