Learning sparse polynomial functions

We study the question of learning a sparse multivariate polynomial over the real domain. In particular, for some unknown polynomial f(x) of degree-d and k monomials, we show how to reconstruct f, within error ε, given only a set of examples xi drawn uniformly from the n-dimensional cube (or an n-dimensional Gaussian distribution), together with evaluations f(xi) on them. The result holds even in the \"noisy setting\", where we have only values f(xi) + g where g is noise (say, modeled as a Gaussian random variable). The runtime of our algorithm is polynomial in n, k, 1/ε and Cd where Cd depends only on d. Note that, in contrast, in the \"boolean version\" of this problem, where x is drawn from the hypercube, the problem is at least as hard as the \"noisy parity problem,\" where we do not know how to break the nΩ(d) time barrier, even for k = 1, and some believe it may be impossible to do so.