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 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.

[1]  A. Zvonkin.  Matrix integrals and map enumeration: An accessible introduction , 1997 .

[2]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[3]  Alexander I. Barvinok,et al.  Estimating L∞ Norms by L2k Norms for Functions on Orbits , 2002, Found. Comput. Math..

[4]  P. Parrilo,et al.  On the equivalence of algebraic approaches to the minimization of forms on the simplex , 2005 .

[5]  Santosh S. Vempala,et al.  The Random Projection Method , 2005, DIMACS Series in Discrete Mathematics and Theoretical Computer Science.

[6]  Erich Kaltofen,et al.  Improved Sparse Multivariate Polynomial Interpolation Algorithms , 1988, ISSAC.

[7]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

[8]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries , 2001, STOC '01.

[9]  Pablo A. Parrilo,et al.  A PTAS for the minimization of polynomials of fixed degree over the simplex , 2006, Theor. Comput. Sci..

[10]  Leonid Gurvits,et al.  Classical complexity and quantum entanglement , 2004, J. Comput. Syst. Sci..

[11]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.

[12]  J. Graver,et al.  Graduate studies in mathematics , 1993 .

[13]  James Renegar.  On the computational complexity and geome-try of the first-order theory of the reals , 1992 .

[14]  M. Brion,et al.  Residue formulae, vector partition functions and lattice points in rational polytopes , 1997 .

[15]  Arieh Iserles,et al.  On the Foundations of Computational Mathematics , 2001 .

[16]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[17]  L. Faybusovich.  Global Optimization of Homogeneous Polynomials on the Simplex and on the Sphere , 2004 .

[18]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.