By Florent Krzakala, Federico Ricci-Tersenghi, Lenka Zdeborova, Riccardo Zecchina, Eric W. Tramel, Leticia F. Cugliandolo
This article gathers the lecture notes of the Les Houches summer time university that used to be held in October 2013 for an viewers of complex graduate scholars and post-doctoral fellows in statistical physics, theoretical physics, desktop studying, and computing device science.
summary: this article gathers the lecture notes of the Les Houches summer season university that was once held in October 2013 for an viewers of complex graduate scholars and post-doctoral fellows in statistical physics, theoretical physics, computer studying, and laptop technology
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Example text
We will now try to determine whether the algorithm can actually reach the fixed points that we have exhibited in the previous section. Let us look at the simple (but loopy) graph in Fig. 16. 80) Understanding belief propagation 21 φ1 1 φ13 φ12 φ2 2 3 φ23 φ3 Fig. 16 φ1 1 φ13 φ12 φ2 3 2 φ3 Fig. 81) x3 which also corresponds to the messages of the modified graph in Fig. 17. 82) x2 corresponding to the messages on the modified graph in Fig. 18. If we increase t, the corresponding non-loopy graph becomes longer at each time step.
Then, finding a junction tree of G is equivalent to finding the max-cut spanning tree of G. 47) e∈T |c1 ∩ c2 | = e∈T 1{v∈e} , = v∈V e∈T and we claim that W (T ) is maximal when T is a JCT. Procedure to obtain the maximal weight spanning tree: • list all edges in decreasing order; • include ei in Ei−1 if possible. At the end of the algorithm we are left with the maximal weight spanning tree. Inference algorithms: elimination, junction tree, and belief propagation 15 Tree width (Definition) The width of a tree decomposition is the size of its maximal clique minus one.
Now we color the two types of vertices in black and white (see above). 9, we define an n × n matrix A. We know already that each perfect matching corresponds to a permutation σ ∈ Sn , which maps each black vertex to its white partner. 9 that perm(A) counts all of these permutations. But the determinant of A counts them weighted by their parities, det(A) = matchings σ (−1)σ . The idea is to compensate the parity weights of the determinant in order to obtain the correct count of perfect matching. To do this, we place weights wij = ±1 on the edges of G.