6,503 research outputs found
On Cavity Approximations for Graphical Models
We reformulate the Cavity Approximation (CA), a class of algorithms recently
introduced for improving the Bethe approximation estimates of marginals in
graphical models. In our new formulation, which allows for the treatment of
multivalued variables, a further generalization to factor graphs with arbitrary
order of interaction factors is explicitly carried out, and a message passing
algorithm that implements the first order correction to the Bethe approximation
is described. Furthermore we investigate an implementation of the CA for
pairwise interactions. In all cases considered we could confirm that CA[k] with
increasing provides a sequence of approximations of markedly increasing
precision. Furthermore in some cases we could also confirm the general
expectation that the approximation of order , whose computational complexity
is has an error that scales as with the size of the
system. We discuss the relation between this approach and some recent
developments in the field.Comment: Extension to factor graphs and comments on related work adde
Towards a four-loop form factor
The four-loop, two-point form factor contains the first non-planar correction
to the lightlike cusp anomalous dimension. This anomalous dimension is a
universal function which appears in many applications. Its planar part in N = 4
SYM is known, in principle, exactly from AdS/CFT and integrability while its
non-planar part has been conjectured to vanish. The integrand of the form
factor of the stress-tensor multiplet in N = 4 SYM including the non-planar
part was obtained in previous work. We parametrise the difficulty of
integrating this integrand. We have obtained a basis of master integrals for
all integrals in the four-loop, two-point class in two ways. First, we computed
an IBP reduction of the integrand of the N = 4 form factor using massive
computer algebra (Reduze). Second, we computed a list of master integrals based
on methods of the Mint package, suitably extended using Macaulay2 / Singular.
The master integrals obtained in both ways are consistent with some minor
exceptions. The second method indicates that the master integrals apply beyond
N = 4 SYM, in particular to QCD. The numerical integration of several of the
master integrals will be reported and remaining obstacles will be outlinedComment: 9 Pages, Radcor/Loopfest 2015 Proceeding
- …