# ISIT 2010: Bethe Entropy and the Edge Zeta Function

Connecting the Bethe Entropy and the Edge Zeta Function of a Cycle Code
by Pascal Vontobel

Pascal talked about a connection of the Bethe entropy to the edge zeta function of a cycle code, building on previous work by Koetter, Li Vontobel and Walker. Starting with the first quantity, as is well known, Yedidia, Freeman and Weiss established that sum-product (if it converges) can only converge to fixed points of the Bethe free energy. The Bethe entropy is the non-linear term added in the ‘average energy’ (optimized by the LP decoder) and is therefore a fundamental quantity to understand sum-product and its connections to LP relaxations. A cycle code is a linear code where all the variables have degree two (and can be represented as edges).

Now what is the second quantity in this connection, the edge zeta function of a cycle code? As far as I understood (and this is probably inaccurate), for each edge (=variable of the code) define an indeterminate quantity $U_{e}$.

For a primitive cycle $c_1$ (without allowing backtracking) of this graph, consisting of edges $e_1, e_2, \cdots$, define a monomial $\gamma(c_1)$ that is the product of terms $\gamma(c_1)= U_{e1}U_{e2}\cdots$ for all the edges of the cycle.

The edge zeta function is a product over all the cycles of these monomials inverted:  $\prod_{\forall c_i} \frac{1}{1-\gamma(c_i)}$.

The edge function can therefore be evaluated at every point of the fundamental polytope by setting the $U_{e}$ appropriately. Previous work by Koetter, Li Vontobel and Walker established that each monomial in the Taylor expansion of this function corresponds to an (unscaled) pseudo-codeword of the cycle code. In particular the powers appearing for each edge (=variable of the code) correspond to re-scaled pseudo-codeword coordinates. As far as I understood, this paper shows that the coefficients of the monomials associated with a given pseudo-codeword have a meaning: their asymptotic growth rate equals to the directional derivative of the (induced) Bethe entropy in the direction of that pseudo-codeword.

I was wondering on the connections to the self-avoiding walks of Dror Weitz [Counting independent sets up to the tree threshold] (also Local approximate inference algorithms)  and the connections of the random walks to the weights of the Arora,Daskalakis, Steurer Message Passing algorithms and improved LP decoding paper.