Regression on High-dimensional Inputs

Consider unknown smooth function which maps high-dimensional inputs, whose values lie on unknown Input manifold of lower dimensionality embedded in an ambient high-dimensional space, to multi-dimensional outputs. Given training dataset consisting of 'input-output' pairs, Regression on input manifold problem is to estimate the unknown function and its Jacobian matrix, as well to estimate the Input manifold. Transforming the high-dimensional inputs to their lowdimensional features, the problem is reduced to certain regression on feature space problem. The paper presents a new geometrically motivated method for solution of both interrelated regression problems.