### abstract ###
We consider computation of permanent of a positive  SYMBOL  non-negative matrix,
  SYMBOL , or equivalently the problem
 of weighted counting of the perfect matchings over the
 complete bipartite graph  SYMBOL
The problem is known to be of likely exponential complexity
Stated as the partition function  SYMBOL  of a graphical model, the problem allows exact Loop Calculus
 representation [Chertkov, Chernyak '06] in terms of an interior minimum of the
 Bethe Free Energy functional over non-integer doubly stochastic matrix of marginal beliefs,  SYMBOL ,
 also correspondent to a fixed point of the iterative message-passing algorithm of the Belief Propagation (BP) type
Our main result is an explicit expression of the exact partition function (permanent) in terms of the matrix of BP marginals,   SYMBOL , as  SYMBOL ,
 where  SYMBOL  is the BP expression for the permanent stated explicitly in terms of  SYMBOL
We give two derivations of the formula, a direct one based on the Bethe Free Energy and
 an alternative one combining the Ihara graph- SYMBOL  function and the Loop Calculus approaches
Assuming that the matrix  SYMBOL  of the Belief Propagation marginals is calculated, we provide two lower bounds and one upper-bound to estimate the multiplicative term
Two complementary lower bounds are based on the Gurvits-van der Waerden theorem and on a relation between the modified permanent and determinant respectively
### introduction ###
The problem of calculating the permanent of a non-negative matrix arises in many contexts in
 statistics,  data analysis and physics
For example, it is intrinsic to the parameter learning of a flow
 used to follow particles in turbulence and to cross-correlate two subsequent images
  CITATION
However, the problem is  SYMBOL -hard   CITATION ,  meaning that solving it
 in a time polynomial in the system size,  SYMBOL , is unlikely
Therefore, when size of the matrix is sufficiently large, one naturally looks for ways to approximate the
 permanent
A very significant breakthrough  was achieved with invention of a so-called
 Fully-Polynomial-Randomized Algorithmic Schemes (FPRAS) for the permanent problem  CITATION :
 the permanent is approximated in a polynomial time, with
 high probability and within an arbitrarily small relative error
However, the complexity of this FPRAS is  SYMBOL , making it impractical for the majority of realistic applications
This motivates the task of finding a lighter deterministic or probabilistic algorithm capable of
 evaluating the permanent more efficiently
This paper continues the thread of  CITATION  and  CITATION , where the Belief Propagation (BP) algorithm was suggested as an efficient heuristic of good (but not absolute) quality to approximate the permanent
The BP family of algorithms,
 originally introduced in the context of error-correction codes  CITATION  and artificial
 intelligence  CITATION , can generally be stated for any graphical model  CITATION
The exactness of the BP on any graph without loops  suggests that the algorithm can be an
 efficient heuristic for evaluating the partition function or for finding a Maximum
 Likelihood (ML) solution for the Graphical Model (GM) defined on sparse graphs
However, in the
 general loopy cases one would normally not expect BP to work well,  thus making the heuristic results of  CITATION  somehow surprising,
 even though not completely unexpected in view of existence of polynomially efficient algorithms for the ML version of the problem  CITATION ,
 also realized in  CITATION  via an iterative BP algorithm
This raises the questions of understanding the performance of BP: what
 it does well and what it misses
It also motivates the challenge of improving the BP
 heuristics
An approach potentially capable of handling the question and the challenge was recently suggested in the general framework of GM
The Loop Series/Calculus (LS) of  CITATION  expresses
 the ratio between the Partition Function (PF) of a binary GM and its BP estimate in terms of a finite series, in which each term is associated with the so-called generalized loop (a subgraph with all vertices of degree larger than one) of the graph
Each term in the series,  as well as the BP estimate of the partition function, is expressed in terms of a doubly stochastic matrix of marginal probabilities,  SYMBOL , for matching pairs to contribute a perfect matching
This matrix  SYMBOL  describes a minimum of the so-called Bethe free energy,  and it can also be understood as a fixed point of an iterative BP algorithm
The first term in the resulting LS is equal to one
Accounting for all the loop-corrections, one recovers the
 exact expression for the PF
In other words,  the LS holds the key to understanding the gap between
 the approximate BP estimate for the PF and the exact result
In section  and section ,
 we will give a technical introduction to the variational Bethe Free Energy (BFE) formulation of BP and a brief
 overview of the LS approach for the permanent problem respectively {Our results }
 In this paper, we develop an LS-based approach to describe the quality of the BP
 approximation for the permanent of a non-negative matrix (i) Our natural starting point is the analysis of the BP solution itself conducted in section
Evaluating the permanent of the non-negative matrix,  SYMBOL ,
 dependent on the temperature parameter,  SYMBOL , we find
 that a non-integer BP solution is observed only at  SYMBOL , where  SYMBOL  is defined by
 \eq() (ii) At  SYMBOL , we derive an alternative representation for the LS in section
The entire LS is collapsed to a product of two terms:
 the first term is an easy-to-calculate function of  SYMBOL , and the second term is the permanent of the matrix,  SYMBOL  (The binary operator  SYMBOL  denotes the element-wise multiplication of matrices )
 This is our main result stated in theorem , and the majority of the consecutive statements of our paper follows from it
We also present yet another, alternative, derivation of the theorem  using the multivariate Ihara-Bass formula for the graph zeta-function in subsection  (iii) Section  presents two easy-to-calculate lower bounds for the LS
The lower bound stated in the corollary  is based on the Gurvits-van der Waerden theorem applied to  SYMBOL
Interestingly enough this lower bound is invariant with respect to the BP transformation,
 i e it is exactly equivalent to the lower bound derived via
 application of the van der Waerden-Gurvits theorem to the original permanent
Another
 lower bound is stated in theorem
Note,  that as follows from an
 example discussed in the text, the two lower bounds are complementary: the latter is stronger
 at sufficiently small temperatures,  while the former
 dominates the large  SYMBOL  region (iv) Section  discusses an upper bound on the transformed permanent based on the application of the
 Godzil-Gutman formula and the Hadamard inequality
Possible future extensions of the approach are discussed in section
