Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace

Abstract. We consider the problem of efficient integration of an n-variate polynomial with respect to the Gaussian measure in Rn 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 fp 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.

• Type: Journal
• Year: 2007
• Where: FOCM
• Authors: Alexander I. Barvinok
• Keywords: FOCM 2007, Integer Power Fp, N-variate Polynomials, Random Subspace