Approximating the a-permanent

The standard matrix permanent is the solution to a number of combinatorial and graph-theoretic problems, and the a-weighted permanent is the density function for a class of Cox processes called boson processes. The exact computation of the ordinary permanent is known to be #P-complete, and the same appears to be the case for the a-permanent for most values of a. At present, the lack of a satisfactory algorithm for approximating the a-permanent is a formidable obstacle to the use of boson processes in applied work. This paper proposes an importance-sampling estimator using nonuniform random permutations generated in a cycle format. Empirical investigation reveals that the estimator works well for the sorts of matrices that arise in point-process applications, involving up to a few hundred points. We conclude with a numerical illustration of the Bayes estimate of the intensity function of a boson point process, which is a ratio of a-permanents.

[1]  V. Vazirani,et al.  Accelerating simulated annealing for the permanent and combinatorial counting problems , 2006, SODA 2006.

[2]  Richard J. Lipton,et al.  A Monte-Carlo Algorithm for Estimating the Permanent , 1993, SIAM J. Comput..

[3]  Richard Sinkhorn.  A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices , 1964 .

[4]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[5]  T. Shirai.  REMARKS ON THE POSITIVITY OF α-DETERMINANTS , 2007 .

[6]  Jun S. Liu,et al.  SEQUENTIAL MONTE CARLO METHODS FOR PERMUTATION TESTS ON TRUNCATED DATA , 2007 .

[7]  P. McCullagh,et al.  Stochastic classification models , 2007 .

[8]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[9]  Ronald L. Graham,et al.  Statistical Problems Involving Permutations With Restricted Positions , 1999 .

[10]  I. Beichl,et al.  Approximating the Permanent via Importance Sampling with Application to the Dimer Covering Problem , 1999 .

[11]  N. Y. Kuznetsov.  Computing the permanent by importance sampling method , 1996 .

[12]  Y. Peres,et al.  Determinantal Processes and Independence , 2005, math/0503110.

[13]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

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

[15]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[16]  Eric Vigoda,et al.  Accelerating simulated annealing for the permanent and combinatorial counting problems , 2006, SODA '06.

[17]  T. Shirai,et al.  Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes , 2003 .

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

[19]  Peter McCullagh,et al.  Stochastic classification models , 2006 .