Learning The Second-Moment Matrix Of A Smooth Function From Point Samples

Consider an open set D subset of R-n, equipped with a probability measure mu. An important characteristic of a smooth function f : D -> R is its second-moment matrix Sigma(mu) := f del f(x)(del f(x))*mu(dx) is an element of R-nxn, where del f (x) is an element of R-n is the gradient of f (.) at x is an element of D. For instance, the span of the leading r eigenvectors of Sigma(mu), forms an active subspace of f (.), thereby extending the concept of principal component analysis to the problem of ridge approximation. In this work, we propose and analyze a simple algorithm for estimating Sigma(mu), from point values of f (.) without imposing any structural assumptions on Sigma(mu).