Learning the Differential Correlation Matrix of a Smooth Function From Point Samples

Consider an open set D ⊆ R, equipped with a probability measure μ. An important characteristic ofa smooth function f : D→ R is its second-moment matrix Σμ :=́ ∇f(x)(∇f(x))∗μ(dx) ∈ Rn×n, where∇f(x) ∈ R is the gradient of f(·) at x ∈ D. For instance, the span of the leading r eigenvectors of Σμforms an active subspace of f(·), thereby extending the concept of principal component analysis to theproblem of ridge approximation. In this work, we propose a simple algorithm for estimating Σμ frompoint values of f(·) without imposing any structural assumptions on f(·). Theoretical guarantees for thisalgorithm are provided with the aid of the same technical tools that have proved valuable in the contextof covariance matrix estimation from partial measurements.