Extended regression on manifolds estimation

Abstract

Let f(X) be unknown smooth function which maps p-dimensional manifold-valued inputs X, whose values lie on unknown Input manifold M of lower dimensionality q < p embedded in an ambient high-dimensional space Rp, to m-dimensional outputs. Regression on manifold problem is to estimate a triple (f(X), Jf(X), M), which includes Jacobian Jf of the mapping f, from given sample consisting of ‘input-output’ pairs. If some mapping h transforms Input manifold M to q-dimensional Feature space Yh = h(M) and satisfies certain conditions, initial estimating problem can be reduced to Regression on feature space problem consisting in estimating of triple (gf(y), Jg,f(y), Yh) in which unknown function gf(y) depends on low-dimensional features y = h(X) and satisfies the condition gf(h(X)) ≈ f(X), and Jg,f is its Jacobian. The paper considers such Extended problem and presents geometrically motivated method for estimating both triples from given sample.