4,199 research outputs found
Approximating the Permanent with Fractional Belief Propagation
We discuss schemes for exact and approximate computations of permanents, and
compare them with each other. Specifically, we analyze the Belief Propagation
(BP) approach and its Fractional Belief Propagation (FBP) generalization for
computing the permanent of a non-negative matrix. Known bounds and conjectures
are verified in experiments, and some new theoretical relations, bounds and
conjectures are proposed. The Fractional Free Energy (FFE) functional is
parameterized by a scalar parameter , where
corresponds to the BP limit and corresponds to the exclusion
principle (but ignoring perfect matching constraints) Mean-Field (MF) limit.
FFE shows monotonicity and continuity with respect to . For every
non-negative matrix, we define its special value to be the
for which the minimum of the -parameterized FFE functional is
equal to the permanent of the matrix, where the lower and upper bounds of the
-interval corresponds to respective bounds for the permanent. Our
experimental analysis suggests that the distribution of varies for
different ensembles but always lies within the interval.
Moreover, for all ensembles considered the behavior of is highly
distinctive, offering an emprirical practical guidance for estimating
permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure
Degree- Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative Matrix
The permanent of a non-negative square matrix can be well approximated by
finding the minimum of the Bethe free energy functions associated with some
suitably defined factor graph; the resulting approximation to the permanent is
called the Bethe permanent. Vontobel gave a combinatorial characterization of
the Bethe permanent via degree- Bethe permanents, which is based on
degree- covers of the underlying factor graph. In this paper, we prove a
degree--Bethe-permanent-based lower bound on the permanent of a non-negative
matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf.
Theory, Mar. 2013]. We also prove a degree--Bethe-permanent-based upper
bound on the permanent of a non-negative matrix. In the limit ,
these lower and upper bounds yield known Bethe-permanent-based lower and upper
bounds on the permanent of a non-negative matrix. Moreover, we prove similar
results for an approximation to the permanent known as the (scaled) Sinkhorn
permanent.Comment: submitte
Loop Calculus for Non-Binary Alphabets using Concepts from Information Geometry
The Bethe approximation is a well-known approximation of the partition
function used in statistical physics. Recently, an equality relating the
partition function and its Bethe approximation was obtained for graphical
models with binary variables by Chertkov and Chernyak. In this equality, the
multiplicative error in the Bethe approximation is represented as a weighted
sum over all generalized loops in the graphical model. In this paper, the
equality is generalized to graphical models with non-binary alphabet using
concepts from information geometry.Comment: 18 pages, 4 figures, submitted to IEEE Trans. Inf. Theor
Approximating the Permanent with Fractional Belief Propagation
We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the belief propagation (BP) approach and its fractional belief propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and Conjectures are verified in experiments, and some new theoretical relations, bounds and Conjectures are proposed. The fractional free energy (FFE) function is parameterized by a scalar parameter y β [β1;1], where y = β1 corresponds to the BP limit and y = 1 corresponds to the exclusion principle (but ignoring perfect matching constraints) mean-field (MF) limit. FFE shows monotonicity and continuity with respect to g. For every non-negative matrix, we define its special value yβ β [β1;0] to be the g for which the minimum of the y-parameterized FFE function is equal to the permanent of the matrix, where the lower and upper bounds of the g-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of yβ varies for different ensembles but yβ always lies within the [β1;β1/2] interval. Moreover, for all ensembles considered, the behavior of yβ is highly distinctive, offering an empirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.Los Alamos National Laboratory (Undergraduate Research Assistant Program)United States. National Nuclear Security Administration (Los Alamos National Laboratory Contract DE C52-06NA25396
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