The Velocity of the Propagating Wave for General Coupled Scalar Systems

0.27 MB·

by  Rafah El-Khatib

The Velocity of the Propagating Wave
for General Coupled Scalar Systems
Rafah El-Khatib and Nicolas Macris
LTHC, EPFL, Lausanne, Switzerland
Emails: {rafah.el-khatib,nicolas.macris}@epfl.ch
Abstract—We consider spatially coupled systems governed by
a set of scalar density evolution equations. Such equations track
the behavior of message-passing algorithms used, for example,
in coding, sparse sensing, or constraint-satisfaction problems.
7
Assuming that the “profile” describing the average state of the
1
algorithmexhibitsasolitonicwave-likebehaviorafterinitialtran-
0
sientiterations,wederiveaformulaforthepropagationvelocity
2
of the wave. We illustrate the formula with two applications,

n namely Generalized LDPC codes and compressive sensing.
a
J I. INTRODUCTION

3
Spatial coupling is a graph construction that was first Fig. 1. The profile x of error probabilities is plotted as a function of the
1
introduced for coding on Low-Density Parity-Check (LDPC) spatialpositionionthecouplingaxisforacoupledGLDPCcode(seesection
IV-A)withn=15,e=3,andchannelnoiseε=0.37.HereL=50andW=4
] codes by Felstrom and Zigangirov [1]. Spatially coupled
(uniformwindow).Weplottheprofileatdifferentiterationsofthemessage-
T
systems have been shown to exhibit excellent performance passingalgorithm.Thesolitontravelingfromlefttorightisplottedevery20
.I under low complexity message-passing algorithms. Due to iterationsuntiliteration180.
s
c this attractive property, they have been extensively studied in
[ different frameworks, such as coding [2], [3] (a review with

applicationsinthebroadercontextofcommunicationsisfound
1
v in[3]),compressivesensing[4],[5],[6],statisticalphysicsand
9 constraint-satisfaction [7], [8], [9], [10].
5 To assess the performance of a message-passing algorithm
7
one analyzes the evolution of the messages exchanged during
3
the algorithm as the number of iterations increases. This can
0
. be expressed as a set of scalar recursive equations (see Equ.
1 (1) for uncoupled systems and Equ. (3) for coupled systems). Fig.2. ThesinglepotentialoftheGLDPCcodeisshownwithn=15,e=3,
0 andwithchannelparameterε=0.37.Noticethatthevaluesofthepositions
7 In coding these are the density evolution equations, and in at which the minima occur match exactly with the boundary values of the
1 compressive sensing these are the state evolution equations. profiles.
: A spatially coupled system is obtained from the underlying
v
i “single” (uncoupled) one by taking 2L+1 copies of it and
X
connecting every W consecutive single systems by means the shape from the initial conditions is left open. The soliton
r of a “coupling window”. For scalar systems, the average isillustratedinFig.1foraGeneralizedLDPC(GLDPC)code
a
behavior of the system at position i ∈ {−L,...,L} of the (see Sec. IV-A for details).
coupling axis and at iteration t ∈N of the message-passing In [11] and [12], bounds on the velocity of the wave for
(t)
algorithm is described by a single scalar x . Therefore, the coding on the BEC are proposed. In [13] a formula for the
i
evolution of the coupled system can be analyzed by tracking velocityofthewaveinthecontextofthecoupledCurie-Weiss
the vector x(t) ={x(t),...,x(t)}, which we call the “profile”, toymodelisderivedandtestednumerically.In[14],aformula
−L L
as t increases. in the context of coding on general binary input memoryless
Under certain initial conditions and after an initial number symmetric channels is derived.
ofiterations,theprofiledemonstratesasolitonicbehavior.That In this work we derive a formula for the velocity of the
is, after a transient phase, it appears to develop a fixed shape wave in the continuum limit L(cid:29)w(cid:29)1 for general scalar
that is independent of the initial condition and travels at a systems. By means of numerical simulations, we find that our
constant velocity as t increases. In fact, it has been proved formula is a very good estimate for the empirical velocity.
in [11] that a solitonic wave solution exists in the context of We limit ourselves to the cases where the scalar recursive DE
codingwhentransmissiontakesplaceovertheBinaryErasure equation of the underlying uncoupled system has exactly two
Channel(BEC).However,thequestionoftheindependenceof stableandoneunstablefixedpoints.Equivalentlythepotential
function (of the uncoupled system) has two minima and one potential function and check that it indeed has 2 stable fixed
local maximum. Fig. 1 illustrates this setting for the GLDPC points (minima) and 1 unstable fixed point (maximum).
example. To obtain the spatially coupled system associated to the
uncoupled one described above, we start by defining the
II. PRELIMINARIES “coupling window” function that satisfies w(z) > 0, when
(cid:82)
A. Density Evolution and Potential Functions 0≤z<1,w(z)=0otherwise,and dzw(z)=1.Then,wede-
R
(cid:16) (cid:17)
We adopt the framework and notations of [15]. Let E = fine the normalized function wW(z)=w(z)/ W1 ∑Wj=−01w(Wj ) .
[a0n,dεmlaext],Xwh=er[e0,εxmmaxax(∈ε)(]0,a∞nd),Yde=no[t0e,ythmaex(εsp)]a,cseucohf tphaartamxmeaxt(eεrs),, Rw(ezm)/ar(cid:82)kRtdhzawt W(1z)∑=Wj=−w01(wz)W. (Wj )=1andthatasW→∞,wW(z)→
ymax(ε)∈(0,∞) and ymax(ε)=g(xmax(ε);ε). Consider bounded Thecoupledsystemisthenobtainedbytaking2L+1copies
and smooth functions f :Y ×E →X and g:X ×E →Y, ofthesinglesystemonthepositionsi={−L,...,L}andcon-
increasing in both arguments. We consider the following necting them using the “coupling matrix” A = 1w (k−j).
j,k W W W
(uncoupled) recursion, The discrete “profile” x(t) = {x(t),...,x(t)} is then fixed at
−L L
x(t+1)= f(g(x(t);ε);ε), (1) the boundaries as follows: x(t)=x , for i={−L,...,−L+
i good
W−1} and all t ∈N, x(t) =x , for i={L−W+1,...,L}
wheret∈Ndenotestheiterationnumber.Therecursionisini- i bad
and all t ∈N. We run density evolution on the remainder of
tializedwithx(0)=x .Since f(g(X))⊂X,theinitialization
max the chain. More specifically, for i∈{−L+W,...,L−W}, the
of the recursion (1) implies that x(1)≤x(0)=x . Moreover,
max coupled scalar recursion is
since the functions f and g are monotonic and bounded, the
recursion will converge to a limiting value x(∞) when the L (cid:16) L (cid:17)
x(t+1)= ∑ A f ∑ A g(x(t);ε);ε , (3)
number of iterations is large, and this limit is a fixed point i j,i j,k k
j=−L k=−L
since f and g are continuous.
The boundary condition here is well adapted to study the
The typical picture in the context of applications such as
propagation of the wave after the transient phase is over.
coding or compressive sensing is as follows. It may help to
Indeed, simulations and heuristic arguments indicate that an
think as ε as the level of noise for coding, as the inverse
initial profile that increases from a seeding value smaller
fraction of measurements in compressed sensing, or as the
or equal to x (at the left boundary) to x (on the right
density of constraints in constraint-satisfaction problems. good max
boundary) is attracted towards the class of profiles defined
We define x as the fixed point of (1) obtained by the
initialization x(g0o)od=0. We furthermore define the algorithmic above.
Thespatiallycoupledsystemcanbedescribedbyapotential
threshold ε as
s
functionalU defined as
c
εs(cid:44)sup{ε|x(∞)=xgood}. L−W (cid:0) (cid:1)
U (x;ε)= ∑ xg(x;ε)−G(x;ε)
c i i i
Since f and g are monotonic, we can see that for ε <ε , the
s i=−L+W
recursion(1)willconvergetoxgood foranyinitializationofx(0) L−W (cid:16) L (cid:17)
in[0,xmax(ε)].Forε>εs newfixedpointsappear.Wewilllimit − ∑ F ∑ Ai,jg(xj;ε);ε , (4)
ourselves to systems with one extra stable fixed point that we i=−L+W j=−L
callxbad suchthatxbad>xgood.Forε>εs theiterationsinitialized where x={x−L,...,xL}.The fixedpointform ofEqu.(3) can
at x converge to x . (It is easy to see that there also must be obtained by setting the derivative with respect to x of the
max bad
exist an extra unstable fixed point x between these two, i.e., potentialU (x;ε) to zero.
unst c
x <x <x .) A highly attractive property of spatially coupled systems
good unst bad
One can equivalently describe the (uncoupled) system by a is that they exhibit the so-called threshold saturation phe-
potential functionUs defined as nomenon. That is, for all ε <εc where εc >εs, the coupled
recursions (3) drive the profile x(t) =[x(t) ,...,x(t)] to the
U (x;ε)=xg(x;ε)−G(x;ε)−F(g(x;ε);ε), (2) −w+1 L
s desirablefixedpointx(∞)=[x ,...,x ].Hereε isathresh-
good good c
where F(x;ε)=(cid:82)xdsf(s;ε) and G(x;ε)=(cid:82)xdsg(s;ε). The old defined by U(x ;ε )=U(x ;ε ) and often called the
0 0 good c bad c
fixed points of (1) can be obtained by setting the derivative potential threshold (note that in this equation x and x
good bad
with respect to x of the potential Us(x;ε) to zero. The stable themselves depend on εc).
fixed points correspond to minima ofU and the unstable one In the sequel we consider the range ε ∈[ε ,ε ]. It is for
s s c
to a local maximum of U . An example of the potential for these values of the parameter ε that a soliton is observed.
s
the GLDPC code (see Sec. IV-A) is shown in Fig. 2. There, Let us repeat our basic assumption here: the recursion (1)
ε >ε and there are two minima corresponding to x =0 has exactly two stable fixed points x and x . With more
s good good bad
and x >0 with one local maximum corresponding to x . than two stable fixed points the propagating wave has a more
bad unst
If we run density evolution, the iterations will get stuck at complicated structure and our formulas would have to be
x >0.Ingeneral,tocheckwhethertheanalysisinthiswork adapted accordingly (see [12] for a nice discussion of this
bad
appliestoacertainapplication,onecanplotitscorresponding issue in coding).
B. Continuum Limit Then, we find the following formula for the velocity v
We consider the system in the continuum limit, which is
U (x ;ε)−U (x ;ε)
obtained by first taking L→∞ and then W →∞ [11], [16], v= (cid:82) sdzgbad(cid:48)(X(z);εs)(Xgo(cid:48)o(dz))2, (6)
[17].Wesetx( i ,t)≡x(t) andreplace i →z, j →u, k →s, R
W i W W W
wherez,u,s∈Rarecontinuousspatialvariables.Thecoupled where g(cid:48)=∂ g is the derivative of g with respect to its first
x
recursion, in the continuum limit, can then be written as argument, and X(cid:48) is the derivative of the profile.
(cid:16) (cid:17)
x(z,t+1)=(cid:82)1duw(u)f (cid:82)1dsw(s)g(x(z−u+s,t);ε);ε . (5)
0 0 B. Derivation of main result
The boundary conditions on the continuous profile x(·,·) Evaluating the functional derivative of ∆W[x(·,·);ε] in an
become x(z,t)→x when z→−∞ and x(z,t)→x when arbitrary direction η(·,·), and then using (5) we obtain
good bad
z→+∞. Again, this boundary condition captures the profiles
obtained after the transient phase has passed, and is well δ∆W[x(·,·);ε]
[η(z,t)]
adapted to the study of the wave propagation. δx(·,·)
Let x (z) be a static (time independent) profile that satisfies 1(cid:110) (cid:111)
0 =lim ∆W[x(·,·)+γη(·,·);ε]−∆W[x(·,·);ε]
the boundary conditions x0(z) → xgood when z → −∞ and γ→0γ
x (z)→x when z→+∞. This profile can be thought as (cid:90) (cid:110)
0 bad = dzη(z,t)g(cid:48)(x(z,t);ε) x(z,t)
an initial condition for the recursions. For us however it R
serves as a reference profile in order to define the potential (cid:90) 1 (cid:90) 1 (cid:111)
− duw(u)f( dsw(s)g(x(z−u+s,t);ε);ε)
functional in the continuum limit. We look at the continuous
0 0
fvreormsiointWof[xU((·)x;;εε])sowthhiacththweeintceagllraWlsc[xo(n·v,e·)r;gεe].Tahnedpsoutbentrtaiaclt =−(cid:90) dzη(z,t)g(cid:48)(x(z,t);ε)(cid:0)x(z,t+1)−x(z,t)(cid:1)
0 R
functional ∆W[x(·,·);ε] in the continuum limit is thus defined
Using the ansatz x(z,t) = X(z−vt) and the approximation
as
x(z,t +1)−x(z,t) ≈ −vX(cid:48)(z), the functional derivative of
(cid:90) (cid:110)
∆W[x(·,·);ε](cid:44) dz x(z,t)g(x(z,t);ε)−x (z)g(x (z);ε) ∆W[x(·,·);ε] in the special direction η(z,t)=X(cid:48)(z) becomes
0 0
R
(cid:16)(cid:90) 1 (cid:17) δ∆W[x(·,·);ε] (cid:90)
−G(x(z,t);ε)−F duw(u)g(x(z−u,t);ε);ε [X(cid:48)(z)]=v dz(X(cid:48)(z))2g(cid:48)(X(z);ε).
0 δx(·,·) R
(cid:16)(cid:90) 1 (cid:17)(cid:111)
+G(x0(z);ε)+F duw(u)g(x0(z−u);ε);ε . To obtain our formula (6) we separate the left-hand side of
0
the equation above into two contributions
As long as x (z) converges to its limiting values fast enough
0
the integrals over the spatial axis are well defined. We can δWs[X(·);ε][X(cid:48)(z)]+δWi[X(·);ε][X(cid:48)(z)]
furthersplitthecontinuouspotentialfunctionalintotwoparts: δX(·) δX(·)
thesinglepotentialW [x(·,·);ε](thatweobtainsettingw(z)→
s
0)andtheinteractionpotentialW[x(·,·);ε](thatiscausedonly and calculate each term separately. For the first (uncoupled)
i
part we find
by coupling), defined as follows
W [x(·,·);ε](cid:44)(cid:90) dz(cid:110)x(z,t)g(x(z,t);ε)−x (z)g(x (z);ε) δWs[X(·);ε][X(cid:48)(z)]
s R 0 0 δX(·)
(cid:90)
−G(x(z,t);ε)−F(g(x(z,t);ε);ε) = dzX(cid:48)(z)(cid:0)X(z)g(cid:48)(X(z);ε)−g(cid:48)(X(z);ε)f(g(X(z);ε);ε)(cid:1)
(cid:111) R
+G(x0(z);ε)−F(g(x0(z);ε);ε) , (cid:90) d (cid:8) (cid:9)
W[x(·,·);ε](cid:44)(cid:90) dz(cid:110)F(g(x(z,t);ε);ε)+F(g(x (z);ε);ε) = Rdzdz X(z)g(X(z);ε)−G(X(z);ε)−F(g(X(z);ε);ε)
i R 0 (cid:104) (cid:105)+∞
= U (X(z);ε) =U (x ;ε)−U (x ;ε).
(cid:16)(cid:90) 1 (cid:17) s s bad s good
−∞
−F duw(u)g(x(z−u,t);ε);ε
0 For the second (interaction) part we find
(cid:16)(cid:90) 1 (cid:17)(cid:111)
+F duw(u)g(x (z−u);ε);ε .
0 δW[X(·);ε]
0 i [X(cid:48)(z)]
δX(·)
III. VELOCITYFORGENERALSCALARSYSTEMS
(cid:90) d (cid:110)
A. Statement of main result = dz F(g(X(z);ε);ε)
R dz
We assume that after a number of iterations, which we call (cid:0)(cid:90) 1 (cid:1)(cid:111)
−F duw(u)g(X(z−u);ε);ε
the transient phase, the density profile x(·,·) develops a fixed
0
shapewhichwecallX(·)thatmoveswithconstantvelocityv. (cid:104) (cid:0)(cid:90) 1 (cid:1)(cid:105)+∞
= F(g(X(z);ε);ε)−F duw(u)g(X(z−u);ε);ε =0
That is, we make the ansatz x(z,t)=X(z−vt).
0 −∞
IV. APPLICATIONS
0.2
Thegeneralformulaforthevelocityofthesolitonforscalar
Empirical velocity
systems (6) can be applied on several examples. In particular
0.15 Analytical velocity
we recover the results of [14] for standard LDPC codes over
theBEC,aswellasongeneralBinaryMemorylessSymmetric ty
i
(BMS) channels within the scalar Gaussian approximation. In loc 0.1
e
V
this section we provide two more scalar applications, namely
GLDPC codes and compressive sensing. 0.05
The predictions of our formula are compared with the
observed,empiricalvelocityve thatisobtainedbyrunningthe 0 0.35 0.355 0.36 0.365 0.37 0.375 0.38 0.385 0.39
scalarrecursions.Toobtainv ,weplotthediscreteprofilexat Channel parameter epsilon
e
differentiterationsofthescalarrecursionsandfindtheaverage
Fig.3. WeconsideraGLDPCcodewithn=15ande=3,withspatiallength
of ∆z/(W∆I), where ∆z is the spatial difference between the L=250 and uniform coupling window withW =3. We plot the velocities
kinks of the profiles, ∆I is the difference in the number of (normalized by W) vGLDPC and ve as a function of the channel parameter
iterations, andW is the size of the coupling window. ε when ε is between the BP threshold εs=εBP=0.348 and the potential
thresholdεc=εMAP≈0.394.
A. Generalized LDPC Codes
We consider a GLDPC code described as follows: The
0.1
variable (bit) nodes have degree 2 and the check nodes have
Empirical velocity
degreen.Therulesofthechecknodesaregivenbyaprimitive 0.08
BCH code of blocklength n (unlike LDPC codes where they Analytical velocity
areparitychecks).AnattractivepropertyofBCHcodesisthat ty0.06
i
c
they can be designed to correct a chosen number of errors. lo
e
V0.04
For instance, one can design a BCH code so that it corrects
all patterns of at most e erasures on the BEC, and all error
0.02
patternsofweightatmosteontheBinarySymmetricChannel
(BSC). 0
0.16 0.165 0.17 0.175 0.18 0.185 0.19 0.195 0.2 0.205
WeconsidertransmissionontheBECorBSCanddenoteby Measurement ratio delta
ε thechannelparameter.Thedensityevolutionrecursionshave
Fig.4. Weconsiderthecompressivesensingproblemwithsnr=105 and
been derived for both channels, based on a bounded distance Gaussian-Bernoulli prior for the signal components with sparsity parameter
decoderfortheBCHcode[18].Wehaveε =x =y =1, ρ=0.1.WehaveL=250anduniformcouplingwindowwithW=4.Weplot
and for n and e fixed [15], max max max thevelocities(normalizedbyW)vCSandveasafunctionofthemeasurement
fraction δ when δ is between the potential threshold δc=0.157 and δs=
0.208.
f(x;ε)=εx,
n−1(cid:18)n−1(cid:19)
g(x,e,n;ε)= ∑ xi(1−x)n−i−1.
i
i=e 0.5
The formula for the velocity of the soliton appearing in the 0.4
case of the GLDPC codes is found from (6), where the single
0.3
potential of the systemU (·) is given by
GLDPC x)
(CS0.2
U
e x(1−x)
U (x,e,n;ε)= g(x,e,n;ε)− g(cid:48)(x,e,n;ε) 0.1
GLDPC n n δ = 0.220
−εg2(x,e,n;ε). 0 δδ == 00..210780
2 δ = 0.157
-0.1
10-8 10-6 10-4 10-2 100
x
Figure 3 shows the velocities (normalized by W) for the
Fig.5. Weconsiderthecompressivesensingproblemwithsnr=105 and
spatially coupled GLPDC code with n=15 and e=3, when
Gaussian-Bernoulli prior for the signal components with sparsity parameter
the coupling parameters satisfy L=250 and W =3 and we ρ =0.1. We plot the single potential for several values of δ ∈[δc,δs]=
use the uniform coupling window. We plot the velocities for [0.157,0.208]. We can see that when δ >δs the potential has a unique
ε ∈[εs,εc]=[0.348,0.394]. We observe that the formula for tmheinipmotuemnt,iawlhheansδtw=oδms,inthimeraeaanpdpetahresaenneirngflyecgtaipon∆pEoiinst,stfroicrtδlyc<poδsit<ivde,elatnads
the velocity provides a very good estimation of the empirical forδ=δc theenergygapvanishes.
velocity v .
e
B. Compressive Sensing ACKNOWLEDGMENTS
R. E. thanks Tongxin Li for fruitful discussions and Jean
Letsbealength-nsignalvector(wherethecomponentsare
Barbierforgivingherpartofhiscodeoncompressivesensing
i.i.d.copiesofarandomvariableS)whichisacquiredthrough
and for patiently explaining it to her. Part of this work was
m linear measurements. We assume that the measurement
√ done during the authors’ stay at the Institut Henri Poincare´ -
matrixhasi.i.dGaussianelements(0,1/ n).Wecallδ=m/n
Centre E´mile Borel. The authors thank this institution for its
thefixedmeasurementratiowhenn→∞.Therelationbetween
δ and the generic ε used in this paper is ε=δ−1. We assume hospitality and support.
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the velocities for δ ∈[δ ,δ ]=[0.157,0.208]. Similarly, we
c s
observe that the formula for the velocity provides a good
estimation of the empirical velocity v .
e