Learning non-parametric basis independent models from point queries via low-rank methods

We consider the problem of learning multi-ridge functions of the form f(x)=g(Ax)f(x)=g(Ax) from point evaluations of f. We assume that the function f   is defined on an ℓ2ℓ2-ball in RdRd, g   is twice continuously differentiable almost everywhere, and A∈Rk×dA∈Rk×d is a rank k   matrix, where k≪dk≪d. We propose a randomized, polynomial-complexity sampling scheme for estimating such functions. Our theoretical developments leverage recent techniques from low rank matrix recovery, which enables us to derive a polynomial time estimator of the function f   along with uniform approximation guarantees. We prove that our scheme can also be applied for learning functions of the form: f(x)=∑i=1kgi(aiTx), provided f satisfies certain smoothness conditions in a neighborhood around the origin. We also characterize the noise robustness of the scheme. Finally, we present numerical examples to illustrate the theoretical bounds in action.