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

Alexander Barvinok

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arXiv · math/0502298

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

Alexander Barvinok

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Abstract

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.

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BibTeX RIS

Alexander Barvinok. 2005-02-14. Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace. https://arxiv.org/abs/math/0502298

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