1,075 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
Palette-colouring: a belief-propagation approach
We consider a variation of the prototype combinatorial-optimisation problem
known as graph-colouring. Our optimisation goal is to colour the vertices of a
graph with a fixed number of colours, in a way to maximise the number of
different colours present in the set of nearest neighbours of each given
vertex. This problem, which we pictorially call "palette-colouring", has been
recently addressed as a basic example of problem arising in the context of
distributed data storage. Even though it has not been proved to be NP complete,
random search algorithms find the problem hard to solve. Heuristics based on a
naive belief propagation algorithm are observed to work quite well in certain
conditions. In this paper, we build upon the mentioned result, working out the
correct belief propagation algorithm, which needs to take into account the
many-body nature of the constraints present in this problem. This method
improves the naive belief propagation approach, at the cost of increased
computational effort. We also investigate the emergence of a satisfiable to
unsatisfiable "phase transition" as a function of the vertex mean degree, for
different ensembles of sparse random graphs in the large size ("thermodynamic")
limit.Comment: 22 pages, 7 figure
Analysis and Optimization of Aperture Design in Computational Imaging
There is growing interest in the use of coded aperture imaging systems for a
variety of applications. Using an analysis framework based on mutual
information, we examine the fundamental limits of such systems---and the
associated optimum aperture coding---under simple but meaningful propagation
and sensor models. Among other results, we show that when thermal noise
dominates, spectrally-flat masks, which have 50% transmissivity, are optimal,
but that when shot noise dominates, randomly generated masks with lower
transmissivity offer greater performance. We also provide comparisons to
classical pinhole cameras
Iterative Quantization Using Codes On Graphs
We study codes on graphs combined with an iterative message passing algorithm
for quantization. Specifically, we consider the binary erasure quantization
(BEQ) problem which is the dual of the binary erasure channel (BEC) coding
problem. We show that duals of capacity achieving codes for the BEC yield codes
which approach the minimum possible rate for the BEQ. In contrast, low density
parity check codes cannot achieve the minimum rate unless their density grows
at least logarithmically with block length. Furthermore, we show that duals of
efficient iterative decoding algorithms for the BEC yield efficient encoding
algorithms for the BEQ. Hence our results suggest that graphical models may
yield near optimal codes in source coding as well as in channel coding and that
duality plays a key role in such constructions.Comment: 10 page
Belief Propagation Min-Sum Algorithm for Generalized Min-Cost Network Flow
Belief Propagation algorithms are instruments used broadly to solve graphical
model optimization and statistical inference problems. In the general case of a
loopy Graphical Model, Belief Propagation is a heuristic which is quite
successful in practice, even though its empirical success, typically, lacks
theoretical guarantees. This paper extends the short list of special cases
where correctness and/or convergence of a Belief Propagation algorithm is
proven. We generalize formulation of Min-Sum Network Flow problem by relaxing
the flow conservation (balance) constraints and then proving that the Belief
Propagation algorithm converges to the exact result
Hierarchical and High-Girth QC LDPC Codes
We present a general approach to designing capacity-approaching high-girth
low-density parity-check (LDPC) codes that are friendly to hardware
implementation. Our methodology starts by defining a new class of
"hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of
quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC
codes are composed of circulant sub-matrices, those of HQC LDPC codes are
composed of a hierarchy of circulant sub-matrices that are in turn constructed
from circulant sub-matrices, and so on, through some number of levels. We show
how to map any class of codes defined using a protograph into a family of HQC
LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the
degrees of freedom within the family of codes to yield a high-girth HQC LDPC
code. Finally, we discuss how certain characteristics of a code protograph will
lead to inevitable short cycles, and show that these short cycles can be
eliminated using a "squashing" procedure that results in a high-girth QC LDPC
code, although not a hierarchical one. We illustrate our approach with designed
examples of girth-10 QC LDPC codes obtained from protographs of one-sided
spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor
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