12 research outputs found
Properties of spatial coupling in compressed sensing
In this paper we address a series of open questions about the construction of
spatially coupled measurement matrices in compressed sensing. For hardware
implementations one is forced to depart from the limiting regime of parameters
in which the proofs of the so-called threshold saturation work. We investigate
quantitatively the behavior under finite coupling range, the dependence on the
shape of the coupling interaction, and optimization of the so-called seed to
minimize distance from optimality. Our analysis explains some of the properties
observed empirically in previous works and provides new insight on spatially
coupled compressed sensing.Comment: 5 pages, 6 figure
The Space of Solutions of Coupled XORSAT Formulae
The XOR-satisfiability (XORSAT) problem deals with a system of Boolean
variables and clauses. Each clause is a linear Boolean equation (XOR) of a
subset of the variables. A -clause is a clause involving distinct
variables. In the random -XORSAT problem a formula is created by choosing
-clauses uniformly at random from the set of all possible clauses on
variables. The set of solutions of a random formula exhibits various
geometrical transitions as the ratio varies.
We consider a {\em coupled} -XORSAT ensemble, consisting of a chain of
random XORSAT models that are spatially coupled across a finite window along
the chain direction. We observe that the threshold saturation phenomenon takes
place for this ensemble and we characterize various properties of the space of
solutions of such coupled formulae.Comment: Submitted to ISIT 201
On the Convergence Speed of Spatially Coupled LDPC Ensembles
Spatially coupled low-density parity-check codes show an outstanding
performance under the low-complexity belief propagation (BP) decoding
algorithm. They exhibit a peculiar convergence phenomenon above the BP
threshold of the underlying non-coupled ensemble, with a wave-like convergence
propagating through the spatial dimension of the graph, allowing to approach
the MAP threshold. We focus on this particularly interesting regime in between
the BP and MAP thresholds.
On the binary erasure channel, it has been proved that the information
propagates with a constant speed toward the successful decoding solution. We
derive an upper bound on the propagation speed, only depending on the basic
parameters of the spatially coupled code ensemble such as degree distribution
and the coupling factor . We illustrate the convergence speed of different
code ensembles by simulation results, and show how optimizing degree profiles
helps to speed up the convergence.Comment: 11 pages, 6 figure
A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions
Low-density parity-check (LDPC) convolutional codes (or spatially-coupled
codes) were recently shown to approach capacity on the binary erasure channel
(BEC) and binary-input memoryless symmetric channels. The mechanism behind this
spectacular performance is now called threshold saturation via spatial
coupling. This new phenomenon is characterized by the belief-propagation
threshold of the spatially-coupled ensemble increasing to an intrinsic noise
threshold defined by the uncoupled system. In this paper, we present a simple
proof of threshold saturation that applies to a wide class of coupled scalar
recursions. Our approach is based on constructing potential functions for both
the coupled and uncoupled recursions. Our results actually show that the fixed
point of the coupled recursion is essentially determined by the minimum of the
uncoupled potential function and we refer to this phenomenon as Maxwell
saturation. A variety of examples are considered including the
density-evolution equations for: irregular LDPC codes on the BEC, irregular
low-density generator matrix codes on the BEC, a class of generalized LDPC
codes with BCH component codes, the joint iterative decoding of LDPC codes on
intersymbol-interference channels with erasure noise, and the compressed
sensing of random vectors with i.i.d. components.Comment: This article is an extended journal version of arXiv:1204.5703 and
has now been accepted to the IEEE Transactions on Information Theory. This
version adds additional explanation for some details and also corrects a
number of small typo
Dynamics and termination cost of spatially coupled mean-field models
This work is motivated by recent progress in information theory and signal
processing where the so-called `spatially coupled' design of systems leads to
considerably better performance. We address relevant open questions about
spatially coupled systems through the study of a simple Ising model. In
particular, we consider a chain of Curie-Weiss models that are coupled by
interactions up to a certain range. Indeed, it is well known that the pure
(uncoupled) Curie-Weiss model undergoes a first order phase transition driven
by the magnetic field, and furthermore, in the spinodal region such systems are
unable to reach equilibrium in sub-exponential time if initialized in the
metastable state. By contrast, the spatially coupled system is, instead, able
to reach the equilibrium even when initialized to the metastable state. The
equilibrium phase propagates along the chain in the form of a travelling wave.
Here we study the speed of the wave-front and the so-called `termination
cost'--- \textit{i.e.}, the conditions necessary for the propagation to occur.
We reach several interesting conclusions about optimization of the speed and
the cost.Comment: 12 pages, 11 figure