Monte Carlo algorithms for computing $$\alpha $$ α -permanents

Abstract

We consider the computation of the $$\alpha $$?-permanent of a non-negative $$n \times n$$n×n matrix. This appears in a wide variety of real applications in statistics, physics and computer-science. It is well-known that the exact computation is a #P complete problem. This has resulted in a large collection of simulation-based methods, to produce randomized solutions whose complexity is only polynomial in $$n$$n. This paper will review and develop algorithms for both the computation of the permanent $$\alpha =1$$?=1 and $$\alpha >0$$?>0 permanent. In the context of binary $$n \times n$$n×n matrices a variety of Markov chain Monte Carlo (MCMC) computational algorithms have been introduced in the literature whose cost, in order to achieve a given level of accuracy, is $$\mathcal {O}(n^7\log ^4(n))$$O(n7log4(n)); see Bezakova (Faster Markov chain Monte Carlo algorithms for the permanent and binary contingency tables. University of Chicago, Chicago, 2008), Jerrum et al. (J Assoc Comput Mach 51:671---697, 2004). These algorithms use a particular collection of probability distributions, the `ideal' of which, (in some sense) are not known and need to be approximated. In this paper we propose an adaptive sequential Monte Carlo (SMC) algorithm that can both estimate the permanent and the ideal sequence of probabilities on the fly, with little user input. We provide theoretical results associated to the SMC estimate of the permanent, establishing its convergence. We also analyze the relative variance of the estimate, associated to an `ideal' algorithm (related to our algorithm) and not the one we develop, in particular, computating explicit bounds on the relative variance which depend upon $$n$$n. As this analysis is for an ideal algorithm, it gives a lower-bound on the computational cost, in order to achieve an arbitrarily small relative variance; we find that this cost is $$\mathcal {O}(n^4\log ^4(n))$$O(n4log4(n)). For the $$\alpha $$?-permanent, perhaps the gold standard algorithm is the importance sampling algorithm of Kou and McCullagh (Biometrika 96:635---644, 2009); in this paper we develop and compare new algorithms to this method; apriori one expects, due to the weight degeneracy problem, that the method of Kou and McCullagh (Biometrika 96:635---644, 2009) might perform very badly in comparison to the more advanced SMC methods we consider. We also present a statistical application of the $$\alpha $$?-permanent for statistical estimation of boson point process and MCMC methods to fit the associated model to data.