14 research outputs found
Pseudo-Codewords of Cycle Codes via Zeta Functions
Cycle codes are a special case of low- density parity-check (LDPC) codes and as such can be decoded using an iterative message-passing decod- ing algorithm on the associated Tanner graph. The existence of pseudo-codewords is known to cause the decoding algorithm to fail in certain instances. In this paper, we draw a connection between pseudo- codewords of cycle codes and the so-called edge zeta function of the associated normal graph and show how the Newton polyhedron of the zeta function equals the fundamental cone of the code, which plays a crucial role in characterizing the performance of iterative de- coding algorithms
Connections Between Computation Trees and Graph Covers
Connections between graph cover pseudocodewords and computation tree pseudocodewords are investigated with the aim of bridging the gap between the theoretically attractive analysis of graph covers and the more intractable analysis of iterative message-passing algorithms that are intuitively linked to graph covers. Both theoretical results and numerous examples are presented
Some properties of graphs determined by edge zeta functions
AbstractIn 1989, Hashimoto introduced an edge zeta function of a finite graph, which is a generalization of the Ihara zeta function. The edge zeta function is the reciprocal of a polynomial in twice as many indeterminants as edges in the graph and can be computed via a determinant expression. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal. Finally, we present a new example illustrating that the Ihara zeta function cannot necessarily do the same
Characterizations of Pseudo-Codewords of LDPC Codes
An important property of high-performance, low complexity codes is the
existence of highly efficient algorithms for their decoding. Many of the most
efficient, recent graph-based algorithms, e.g. message passing algorithms and
decoding based on linear programming, crucially depend on the efficient
representation of a code in a graphical model. In order to understand the
performance of these algorithms, we argue for the characterization of codes in
terms of a so called fundamental cone in Euclidean space which is a function of
a given parity check matrix of a code, rather than of the code itself. We give
a number of properties of this fundamental cone derived from its connection to
unramified covers of the graphical models on which the decoding algorithms
operate. For the class of cycle codes, these developments naturally lead to a
characterization of the fundamental polytope as the Newton polytope of the
Hashimoto edge zeta function of the underlying graph.Comment: Submitted, August 200
Belief Propagation and Loop Series on Planar Graphs
We discuss a generic model of Bayesian inference with binary variables
defined on edges of a planar graph. The Loop Calculus approach of [1, 2] is
used to evaluate the resulting series expansion for the partition function. We
show that, for planar graphs, truncating the series at single-connected loops
reduces, via a map reminiscent of the Fisher transformation [3], to evaluating
the partition function of the dimer matching model on an auxiliary planar
graph. Thus, the truncated series can be easily re-summed, using the Pfaffian
formula of Kasteleyn [4]. This allows to identify a big class of
computationally tractable planar models reducible to a dimer model via the
Belief Propagation (gauge) transformation. The Pfaffian representation can also
be extended to the full Loop Series, in which case the expansion becomes a sum
of Pfaffian contributions, each associated with dimer matchings on an extension
to a subgraph of the original graph. Algorithmic consequences of the Pfaffian
representation, as well as relations to quantum and non-planar models, are
discussed.Comment: Accepted for publication in Journal of Statistical Mechanics: theory
and experimen
Characterizations of pseudo-codewords of LDPC codes
An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these algorithms, we argue for the characterization of codes in terms of a so called fundamental cone in Euclidean space which is a function of a given parity check matrix of a code, rather than of the code itself. We give a number of properties of this fundamental cone derived from its connection to unramified covers of the graphical models on which the decoding algorithms operate. For the class of cycle codes, these developments naturally lead to a characterization of the fundamental polytope as the Newton polytope of the Hashimoto edge zeta function of the underlying graph