Integration and optimization of multivariate polynomials by restriction onto a random subspace

We consider the problem of efficient integration of an n-variate polynomial with respect to the Gaussian measure in R-n and related problems of complex integration and optimization of a polynomial on the unit sphere. We identify a class of n-variate polynomials f for which the integral of any positive integer power f P over the whole space is well approximated by a properly scaled integral over a random subspace of dimension O (log n). Consequently, the maximum of f on the unit sphere is well approximated by a properly scaled maximum on the unit sphere in a random subspace of dimension O (log n). We discuss connections with problems of combinatorial counting and applications to efficient approximation of a hafnian of a positive matrix.