33,384 research outputs found
Planar Graphical Models which are Easy
We describe a rich family of binary variables statistical mechanics models on
a given planar graph which are equivalent to Gaussian Grassmann Graphical
models (free fermions) defined on the same graph. Calculation of the partition
function (weighted counting) for such a model is easy (of polynomial
complexity) as reducible to evaluation of a Pfaffian of a matrix of size equal
to twice the number of edges in the graph. In particular, this approach touches
upon Holographic Algorithms of Valiant and utilizes the Gauge Transformations
discussed in our previous works.Comment: 27 pages, 11 figures; misprints correcte
General duality for abelian-group-valued statistical-mechanics models
We introduce a general class of statistical-mechanics models, taking values
in an abelian group, which includes examples of both spin and gauge models,
both ordered and disordered. The model is described by a set of ``variables''
and a set of ``interactions''. A Gibbs factor is associated to each variable
and to each interaction. We introduce a duality transformation for systems in
this class. The duality exchanges the abelian group with its dual, the Gibbs
factors with their Fourier transforms, and the interactions with the variables.
High (low) couplings in the interaction terms are mapped into low (high)
couplings in the one-body terms. The idea is that our class of systems extends
the one for which the classical procedure 'a la Kramers and Wannier holds, up
to include randomness into the pattern of interaction. We introduce and study
some physical examples: a random Gaussian Model, a random Potts-like model, and
a random variant of discrete scalar QED. We shortly describe the consequence of
duality for each example.Comment: 26 pages, 2 Postscript figure
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Kinematic analysis of quick-return mechanism in three various approaches
Članak se bavi kinematičkom analizom Whitwordovog mehanizma koja je izvedena u tri različite metode. Suvremene metode su računalno potpomognute s posebnim softverom za analizu obrade, koji može simulirati ne samo gibanje mehanizma, već može odrediti i položaj, brzinu, ubrzanje, sile, momente te druge parametre u svakom trenutku vremena, ali je potrebna provjera i razumijevanje zakona mehanike. Cilj je kinematičke analize istražiti gibanje pojedinih komponenti mehanizma (ili njegovih važnih točaka) u ovisnosti o gibanju pobuđivača. Ovdje su opisani osnovni principi triju pristupa te prednosti i nedostaci prezentiranih rješenja. Dobiveni se rezultati mogu usporediti, ako su rabljeni isti ulazni parametri.The article deals with kinematic analysis of quick-return mechanism that is executed by three various methods. The modern methods are computer aided with the special software for analysis processing, which can simulate not only the motion of the mechanism, but can define the position, velocity, acceleration, forces, moments and other parameters at every moment of time, but verification and mechanics laws understanding are necessary. The goal of the kinematic analysis is to investigate the motion of individual components of mechanism (or its important points) in dependence on the motion of drivers. The article describes the basic principles of three approaches, as well as the advantages and disadvantages of presented solutions. The obtained results can be compared, if the same input parameters are used
Normal Factor Graphs and Holographic Transformations
This paper stands at the intersection of two distinct lines of research. One
line is "holographic algorithms," a powerful approach introduced by Valiant for
solving various counting problems in computer science; the other is "normal
factor graphs," an elegant framework proposed by Forney for representing codes
defined on graphs. We introduce the notion of holographic transformations for
normal factor graphs, and establish a very general theorem, called the
generalized Holant theorem, which relates a normal factor graph to its
holographic transformation. We show that the generalized Holant theorem on the
one hand underlies the principle of holographic algorithms, and on the other
hand reduces to a general duality theorem for normal factor graphs, a special
case of which was first proved by Forney. In the course of our development, we
formalize a new semantics for normal factor graphs, which highlights various
linear algebraic properties that potentially enable the use of normal factor
graphs as a linear algebraic tool.Comment: To appear IEEE Trans. Inform. Theor
Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs
We present an asymptotic expansion for quaternionic self-adjoint matrix
integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon
graphs and their non-orientable counterparts. The result exhibits a striking
duality between quaternionic self-adjoint and real symmetric matrix integrals.
The asymptotic expansions of these integrals are given in terms of summations
over topologies of compact surfaces, both orientable and non-orientable, for
all genera and an arbitrary positive number of marked points on them. We show
that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble
(GSE) have exactly the same graphical expansion term by term (when
appropriately normalized),except that the contributions from non-orientable
surfaces with odd Euler characteristic carry the opposite sign. As an
application, we give a new topological proof of the known duality for
correlations of characteristic polynomials. Indeed, we show that this duality
is equivalent to Poincare duality of graphs drawn on a compact surface. Another
application of our graphical expansion formula is a simple and simultaneous
(re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary
Ensemble) and GSE: The three cases have exactly the same graphical limiting
formula except for an overall constant that represents the type of the
ensemble.Comment: 39 pages, AMS LaTeX, 49 .eps figures, references update
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