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.
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Barvinok, A. Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace. Found Comput Math 7, 229–244 (2007). https://doi.org/10.1007/s10208-005-0178-x
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DOI: https://doi.org/10.1007/s10208-005-0178-x