Statistical Learning via Manifold Learning

A new geometrically motivated method is proposed for solving the non-linear regression task consisting in constructing a predictive function which estimates an unknown smooth mapping f from q-dimensional inputs to m-dimensional outputs based on a given ´input-output´ training pairs. The unknown mapping f determines q-dimensional Regression manifold M(f) consisting of all the (q+m)-dimensional ´input-output´ vectors. The manifold is covered by a single chart, the training data set determines a manifold-valued sample from this manifold. Modern Manifold Learning technique is used for constructing the certain estimator M* of the Regression manifold from the sample which accurately approximates the Regression manifold. The proposed method called Manifold Learning Regression (MLR) finds the predictive function fMLR to ensure an equality M(fMLR) = M*. The MLR estimates also the m×q Jacobian matrix of the mapping f.