Analysis of Coupled Scalar Systems by Displacement Convexity

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by  Rafah El-Khatib

Analysis of Coupled Scalar Systems by
Displacement Convexity
Rafah El-Khatib∗, Nicolas Macris∗, Tom Richardson† and Ruediger Urbanke∗
∗School of Computer and Communication Sciences
EPFL, Lausanne, Switzerland
Emails: {rafah.el-khatib,nicolas.macris,ruediger.urbanke}@epfl.ch
† Qualcomm, USA
Email: [email protected]
Abstract—Potential functionals have been introduced recently under rather mild assumptions. For this purpose we will use
as an important tool for the analysis of coupled scalar systems thepotentialintherepresentationof[1]whichallowstoobtain
(e.g. density evolution equations). In this contribution we inves-
more transparent, general, and simpler proofs.
7 tigate interesting properties of this potential. Using the tool of
1 displacement convexity we show that, under mild assumptions This manuscript is organized as follows. Section II intro-
0 on the system, the potential functional is displacement convex. ducesthemodelandthevariationalformulation.InSectionIII
2 Furthermore, we give the conditions on the system such that we proverearrangementinequalitiesthatallow us to naturally
n the potential is strictly displacement convex in which case the considerthepotentialasafunctionalofcumulativedistribution
a minimizer is unique. functions. The potential is shown to be displacement convex
J
3 I. INTRODUCTION inSectionIV.Strictdisplacementconvexityandunicityofthe
minimizer are proved in Section V.
1
Spatially coupled systems have been used recently in

various frameworks, such as compressive sensing, statistical
] II. SETUP AND VARIATIONALFORMULATION
T physics, coding, and random constraint satisfaction problems
I (see [1] and the references therein for a review of the litera- The natural setting for displacement convexity is the con-
.
s ture).Theyhavebeenshowntoexhibitexcellentperformance, tinuum case which can be thought of as an approximation of
c
often optimal, under low complexity message passing algo- the discrete system in the regime of large spatial length and
[
rithms. For example,spatially coupledcodesachieve capacity windowsize.Thecontinuumlimithasalreadybeenintroduced

1
under such algorithms [1]. intheliteratureasaconvenientmeanstoanalyzethebehavior
v
The performance of these systems is assessed by the solu- of the original discrete model [1], [3], [8], [9].
7
6 tionsof(coupled)DensityEvolution(DE)updateequations.In Consider a spatially coupled system with an averaging
7 general, these equationscan be viewed as the stationary point window w : R → R which is always assumed to be
3 equationsofafunctionalthatistypicallycalled“thepotential”. bounded, non negative, even, integrable and normalized such
0
Ithasalreadybeenrecognizedthatthisvariationalformulation that dxw(x) = 1 (as we will see, sometimes further
. R
1 is apowerfultoolto analyzeDE updatesundersuitable initial assumptions will be necessary depending on the statements).
0 conditions[2].Therearevariouspossibleformulationsofthis We dRenote by ⊗ the standard convolution on R and express
7
potentialfunctional[1], [2], [3], [4]. In this paperwe will use the “fixed point DE equations of a scalar continuous system”
1
the representation of [1] for scalar systems. as follows:
:
v
In a previous contribution [5], [6] we showed that the
i
X potential (in the form [2]) associated to a spatially coupled g(x)=hg((f ⊗w)(x)), (1)
r Low-DensityParity-Check(LDPC) codewhose single system f(x)=hf((g⊗w)(x)), (2)
a is the (ℓ,r)-regular Gallager ensemble, with transmission
over the BEC(ǫ), has a convex structure called “displacement where x ∈ R is the spatial position. We will often use the
convexity”. This structure is well known in the theory of shorthand notation fw =f ⊗w and gw =g⊗w; further, we
optimal transport [7]. In fact the potential we consider in [5], will often refer to the functionsf, g as profiles and to h , h
f g
[6] is not convex in the usual sense but it is in the sense of as update functions.
displacementconvexity.Thisinitselfisaninterestingproperty. We will also adopt a convenientnormalizationfor all these
When displacement convexity is strict one deduces that the functions. As will become clear in the example below it
minimum of the potential is unique — assuming it exists — is always possible to adopt this normalization in specific
and thus so is the solution of the DE equation. applications. First, we assume that the profiles are bounded.
The main purpose of the present note is to prove that a Specifically,f,g :R→[0,1].Next,weassumethattheupdate
general class of scalar systems also exhibits the property of functions h and h are non-decreasing bounded functions
f g
displacementconvexity,andevenstrictdisplacementconvexity h :[0,1]→[0,1] normalized such that h (0)=h (0)=0
f,g f g
profiles f and g,
h (x)
f
0.8 W(f,g)= dx I (x)−f(x)gw(x) (4)
f,g
RZ n o
0.6
where
h−1(x)
g g(x) f(x)
0.4 I (x)= duh−1(u)+ dvh−1(v).
f,g g f
Z0 Z0
In order for the integral over R in (4) to exist we have to
0.2
impose some conditions on the profiles f and g. These will
thus be taken in the spaces (here ǫ>0)
x
0
0 0.2 0.4 0.6 0.8 S ={f :R→[0,1], (5)
f
lim x1+ǫf(x)=0, lim x1+ǫ(f(x)−1)=0},
Fig. 1. Anexample ofthe systems weconsider. TheEXIT-like curves are x→−∞ x→+∞
hf (in red) and h−g1 (in blue). The signed area A(hf,hg;1) from (3) is S ={g :R→[0,1], (6)
the sum of the light gray areas (positively signed) and the dark gray areas g
(negatively signed), anditisequalto0. lim x1+ǫg(x)=0, lim x1+ǫ(g(x)−1)=0}.
x→−∞ x→+∞
The fixed point profile solutions of the DE equations (1), (2)
and h (1) = h (1) = 1. We will think of them as EXIT- canbeseenastheleftsideofthe(symmetric)decodingwaves.
f g
like curves(u,h (u)) and (h (v),v) for u,v ∈[0,1] (see the In this paper we assume that these solutions belong to the
f g
generic plot). spacesS andS .Thisisachievedundersomemildconditions
f g
Consider the signed area between the two curves, namely on the slopes of the update functions hf and hg at the corner
points, and if w decays fast enough. In particular this is true
u
A(h ,h ;u)= du′(h−1(u′)−h (u′)). (3) for the example of the BEC(ℓ,r) with a finitely supported w,
f g g f
Z0 wherethelimitingvaluesareapproachedatleastexponentially
fast.
This is a functional of h , h and a function of u ∈ [0,1].
f g
We considerthe case where A(hf,hg;u)>0 for allu∈]0,1[ III. REARRANGEMENTS
andA(h ,h ;1)=0.Thisisequivalenttothestrictlypositive
f g
Displacementconvexityisdefinedonaspaceofprobability
gap condition of [1]. In [1] the condition was stated in terms
measures. For measures on the real line it is most convenient
of the function of u and v
to view displacement convexity on a space of cumulative
u v
φ(h ,h ;u,v)= du′h−1(u′)+ dv′h−1(v′) −uv. distributionfunctions(cdf’s).Inthissectionweusethetoolof
f g g f increasing rearrangements to show that such rearrangements
Z0 Z0
of f and g can only decrease the potential.
theconditionbeingthatφispositivefor(u,v)∈[0,1]2except
We first give a brief introduction to the notion of non-
at (0,0) and (1,1) where it takes the value 0. A(hf,hg;u) is decreasingrearrangement,see[10].Consideraprofilep:R→
obtained by minimizing φ(h ,h ;u,v) over v.
f g [0,1] such that lim p(x) = 0 and lim p(x) = 1. Then,
x→−∞ x→+∞
Example Take a spatially coupled LDPC code whose single the increasing rearrangement1 of p is the non-decreasing
system is the (ℓ,r)-regular Gallager ensemble, with trans- function p¯ so that this function has the same limits and
mission over the BEC(ǫ). Let u (resp. v) be the erasure the mass of each level set is preserved. More formally, let
probabilities emitted by check (resp. variable) nodes. Let us represent p in layer cake form as p(x) = p(x)dt =
the functions h˜f (resp. h˜g) give the usual erasure probabil- +∞dtχ (x), where χ (x) is the indicator functi0on of the
ities emitted on variable (resp. check) node sides. Explicitly, 0 t t R
level set E = {x|p(x) > t}. For each value t ∈ [0,+∞[,
h˜ (u) = ǫuℓ−1 and h˜ (v) = 1−(1−v)r−1, and the usual R t
f g the level set E can be written as the union of a bounded
DE equations are v = h˜ (u), u = h˜ (v). We consider the t
f g set A and a half line ]a ,+∞[. We define the rearranged set
t t
ffisupxneeccditaioplnocsianshtesǫua=cnhdǫMthhAPa.taLrAeet(h˜d(ueffiM,An˜hPe,gdv;uMaAMsPA)Ph)be(=uth)e0=.unTh˜ihqeu(eunnoornmu-)at/rlivivzieadl ET¯hten=,]pa¯(tx−)=|At0|+,+∞∞dt[χ¯atn(dx)χ.¯t the indicator function of E¯t.
f g f f MAP MAP Proposition 3.1: Assume that the window function w is
and hg(v) = h˜g(vMAPv)/uMAP. The coupled DE fixed point symmetric decRreasing2. Let f and g be in Sf and Sg re-
equations are spectively, and let f¯ and g¯ be their respective increasing
u g(x)=1− 1−fw(x)v r−1, f(x)= gw(x) ℓ−1. rearrangements. Then, W(f,g)≥W(f¯,g¯).
MAP MAP
It is not diffic(cid:0)ult to check th(cid:1)at the DE equa(cid:0)tions ((cid:1)1)-(2) 1Notethatanincreasingrearrangementisnotnecessarilystrictlyincreasing.
2We say that a function is symmetric decreasing if it is even and non-
are stationarypointequationsofa potentialfunctionalofboth increasing onthepositive halfline.
To prove Proposition 3.1, we make use of the Riesz rear- Now consider fˆ∗ and gˆ∗ the symmetric decreasing rearrange-
rangement inequality in one dimension [11]. ments of fˆ, and gˆ, respectively.Each of the above terms may
Let p be a non-negative measurable function on R. Then only decrease upon rearrangements. Indeed each of the first
the symmetric decreasingrearrangement p∗ of p is definedas two terms is a functional of a single monotone function and
an even function, that is decreasing on [0,+∞[ and such that thus remains unchanged by rearrangement: U (gˆ) = U (gˆ∗)
1 1
the level sets {x|p(x) ≥ t} and {x|p∗(x) ≥ t} have equal and U (fˆ) = U (fˆ∗) (see for example [11] p. 80 (3.3)).
2 2
Lebesgue measure. Note that the decrease on [0,+∞[ is not The term (10) decreases upon rearrangement as a direct
necessarily strict. application4 of Lemma 3.2. We thus conclude that
Lemma 3.2 (Riesz’s Inequality): Let f , f , and f be any
1 2 3 1
measurablenon-negativefunctionsontherealline,andf∗,f∗, W(fˆ,gˆ)≥ lim W(fˆ∗,gˆ∗).
and f∗ be their symmetric decreasing rearrangements.T1hen23, 2R→+∞
3
To obtain W(f,g) ≥ W(f¯,g¯) it remains to remark that
R2dxdyf1(x)f2(x−y)f3(y) 21limR→+∞W(fˆ∗,gˆ∗) = W(f¯,g¯). This is achieved by re-
ZZ versing the steps (7)-(9).
≤ dxdyf∗(x)f∗(x−y)f∗(y). From now on we therefore restrict the functional to the
1 2 3
ZZR2 spaces of non-increasing profiles.
Proofof Proposition3.1: Consider the expressionof the
potential in (4). In order to make use of the Riesz inequality IV. DISPLACEMENT CONVEXITY
we first “symmetrize” the profiles f and g, and rewrite the
A generic functional F(p) on a space X (of “profiles”
functionalin termsof symmetricprofiles. Choose R>0 very
say) is said to be convex in the usual sense if for any pair
large but fixed. Eventually we will take R → +∞. Denote
p ,p ∈ X, and for all λ ∈ [0,1], the inequality F((1 −
by fˆthe profile such that fˆ(x) = f(x), x < R and fˆ(x) = 0 1
λ)p +λp ) ≤ (1−λ)F(p )+λF(p ) holds. Displacement
fˆ(2R−x), x > R, and by gˆ the function such that gˆ(x) = 0 1 0 1
convexity, on the other hand, is defined as convexity under
g(x), x<R and gˆ(x)=gˆ(2R−x), x>R.
an alternative interpolation called displacement interpolation.
We now write the potential in (4) in terms of the sym-
The right setting for displacement convexity is a space of
metrized profiles fˆand gˆ. We have
probabilitymeasures.For measuresover the realline one can
R conveniently define the displacement interpolation in terms
W(f,g)= lim dx I (x)−f(x)gw(x)
f,g of the cdf’s associated to the measures. This is the simplest
R→+∞Z−∞ n o setting and the one that we adopt here.
R R
= lim dxI (x)− dxf(x)gw(x) . (7) Wethinkoftheincreasingprofilesf andg ascdf’sofsome
f,g
R→+∞(cid:26)Z−∞ Z−∞ (cid:27) underlying measures over the real line. Consider two pairs
For the firstterm in the bracketsaboveitis straightforwardto (f ,g ) and (f ,g ), and define two (pushforward) maps T
0 0 1 1 f
see that and T as
g
R 1
dxI (x)= dxI (x). (8) T (x)=f−1(f (x)), T (x)=g−1(g (x)).
Z−∞ f,g 2ZR fˆ,gˆ f 1 0 g 1 0
For the second term slight care must be taken because of the Consider the linear interpolation between points on R,
convolution. We find
x =(1−λ)x+λT (x), x =(1−λ)x+λT (x).
f,λ f g,λ g
R
dx dyf(x)w(x−y)g(y) (9) The displacement interpolants(f ,g ) are defined so that the
λ λ
Z−∞ ZR following equalities hold for all λ∈[0,1] and x∈R
1 1
= dx dyfˆ(x)w(x−y)gˆ(y)+o( ).
2ZR ZR R2 fλ(xf,λ)=f0(x), gλ(xg,λ)=g0(x).
Note that fˆand gˆ are integrable over R. Then we can write We now state the main result of this section.
W(f,g)= 1 lim (U (gˆ)+U (fˆ)+U (fˆ,gˆ)) Proposition 4.1: The potential W(f,g) is displacement
2R→+∞ 1 2 3 convex which means that for all λ∈[0,1]
where
W(f ,g )≤(1−λ)W(f ,g )+λW(f ,g ).
λ λ 0 0 1 1
gˆ(x)
U1(gˆ)= dx duh−g1(u), We define the two following quantities
ZR Z0
fˆ(x) x x
U (fˆ)= dx dvh−1(v), Ω(x)= dzw(z), V(x)= dzΩ(z), (11)
2 f
ZR Z0 Z−∞ Z−∞
U (fˆ,gˆ)=− dx dyfˆ(x)w(x−y)gˆ(y). (10) and call V the kernel for reasons which will appear shortly.
3
R R
Z Z 4We apply this inequality for f1 = f, f2 = w, f3 = g. As assumed in
3Ifthelefthandsideisinfinitesoistherighthandsideandtheinequality Proposition 3.1 w is symmetric decreasing window so that for us w(x) =
issatisfied. w∗(x)forallx∈R.
Before proving the proposition let us first note In the last line we used a change of variables u = g (x).
0
Using a similar analysis for the other terms in (13) we find
dx(f(x)−f(+∞))gw(x)= dx(fw(x)−f(+∞))g(x) W (f ,g )−W (f ,g )
R R 1 λ λ 1 0 0
Z Z
= dx dy(f(y)−f(+∞))w(y−x)g(x) =λ dg0(x)(x−Tg(x))h−g1(g0(x))
ZR ZR ZR
=− df(y)V(y−x)dg(x),
ZRZR +df0(x)(x−Tf(x))h−f1(f0(x))+dg0(x)(x−Tg(x)) .
!
where we have used integration by parts and g(−∞) =
f(−∞)=0 for the last step. WethusconcludethatW1(fλ,gλ)islinear,andhenceconvex,
Proof of Proposition 4.1: Using the last identity we in λ.
rewrite the potential in (4) as follows. WenowconsiderthedoubleintegraltermW2(f,g)in(12).
Using again a change of variables write
g(x) f(x)
W(f,g)= dx duh−1(u)+ dvh−1(v)
RZ (Z0 g Z0 f W2(fλ,gλ)=ZZR2dfλ(y)V(y−x)dgλ(x)
= df (y)V (1−λ)(y−x)
−f(+∞)g(x) + df(y)V(y−x)dg(x). 0
R2
) R2 ZZ (cid:16)
ZZ
+λ(T (y)−T (x)) dg (x).
(12) f g 0
(cid:17)
We now express the potential as the sum W(f,g) = This is convex in λ because the kernel V is (see (11)).
W (f,g) + W (f,g), where W (f,g) consists of the last
1 2 2 V. UNICITY OFMINIMIZER
double integral in (12).
In this section we prove that the potential is strictly dis-
We first consider W (f,g) and write
1
placement convex under the strictly positive gap condition.
gλ(x) This implies that it admits a unique minimizer.
W (f ,g )−W (f ,g )= dx duh−1(u) (13)
1 λ λ 1 0 0 g Under this condition and assuming that w is even and
ZR (Z0
regular5, the existence of increasing fixed point solutions was
g0(x) fλ(x) f0(x)
− duh−1(u)+ dvh−1(v)− dvh−1(v) established in [1]. It was also shown in [1] that existence
g f f
Z0 Z0 Z0 of such a fixed point implies a positive gap condition6. It
− fλ(+∞)gλ(x)+f0(+∞)g0(x) . was shown that if A(hf,hg;u) = 0 for some u ∈]0,1[ then
) there may be an infinite family of fixed point solutions not
(cid:16) (cid:17)
equivalentundertranslation.Theproofofunicityreliesonthe
We remark that
potential function formulationso the regularity conditions(5)
dx gλ(x)duh−1(u)− g0(x)duh−1(u) and(6)arerequired.Theycanbeshowntobenecessaryunder
g g
ZR Z0 Z0 ! mild assumptionson the scalar recursionstability of the fixed
1 points (0,0) and (1,1) and on the decay of w.
= dx du Θ(g (x)−u)−Θ(g (x)−u) h−1(u),
λ 0 g Fromtheresultsinthepreceedingsectionitfollowsthatall
ZR Z0 (cid:16) (cid:17) (14) increasing fixed points must have the same potential and this
potential is minimal.
where Θ is the Heaviside step function. One can check by
Let f ,g and f ,g both be non-decreasing fixed points.
considering the two cases g (x) > g (x) and g (x) > g (x) 0 0 1 1
λ 0 0 λ
We claim that they must be translates of each other, i.e., y−
that
T (y)is constantdf -almosteverywhere(a.e.)andx−T (x)
f 0 g
1
takes the same constant value dg -a.e. Note that one of these
du Θ(g (x)−u)−Θ(g (x)−u) =|g (x)−g (x)|. 0
λ 0 λ 0
conditions implies the other since both pairs are fixed points.
Z0 (cid:12) (cid:12)
This obs(cid:12)ervation allows us to use Fubi(cid:12)ni’s theorem to swap We will show that y−Tf(y) is constantdf0-a.e. The method
(cid:12) (cid:12)
ofproofistoshowthatif thisisnotthecase thenW (f ,g )
the integrals in (14). We then write (14) as 2 λ λ
is strictly convex at λ = 0 which contradicts the minimality
1 of f ,g .
duh−1(u) dx Θ(g (x)−u)−Θ(g (x)−u) 1 1
g λ 0 The proof relies on results from [1] that relate the strictly
Z0 1 ZR (cid:16) (cid:17) positive gap condition to the positivity of certain integrals of
= duh−g1(u) dx Θ(x−gλ−1(u))−Θ(x−g0−1(u)) spatial fixed points, which will be shown to imply the strict
Z01 ZR (cid:16) (cid:17) convexityof W2(fλ,gλ) if y−Tf(y) is not constant df0-a.e.
= duh−1(u)(g−1(u)−g−1(u))
g 0 λ 5Regularityofwmeansthatitisstrictlypositiveonaninterval(−W,W),
Z0
W ≤+∞and0offof[−W,W].
=λ dg0(x)(x−Tg(x))h−g1(g0(x)) 6ThestrictlypositivegapconditionrequiresthatA(hf,hg;u)>0whereas
ZR thepositive gapcondition requires onlythatA(hf,hg;u)≥0.
Since we have no further need for explicit use of f ,g we x+ ∈ A+, and x− ∈ A−, such that |x − x+| < W and
1 1 f f f f g f
will simplify notation and refer to f0,g0 as f,g. |xg −x−f|<W.
Let us introduce the following functional from [1], Assume now the existence of an s so that A+ and A− are
f f
bothnon-empty.Wewillshowthatthisimpliesstrictconvexity
ξ (w;f,g;x ,x )= (15)
φ 1 2 byestablishingtheexistenceofanx asabove.Sincethisisa
+∞ g
contradiction we conclude that x−T (x) is constant df-a.e.,
dxw(x) df(y)(g(x +)−g(y−x)) f
1
Z0 Z]x2,x1+x] giving the desired result.
+∞ If there exists z ∈ A− ∩A+ then we take x+ = x− = z.
f f f f
+ dxw(x) dg(y)(f(x +)−f(y−x))
2 Sincethestrictlypositivegapconditionimpliesdg(z−W,z+
Z0 Z]x1,x2+x] W) > 0 (by (16)) we have A ∩(z −W,z +W) 6= ∅ and
g
where the integrals are Lebesgue-Stieltjes integrals. If x′ <x we can find a suitable x . Assume now that A+ ∩A− = ∅.
g f f
then ]x,x′]dg(y)f(y) is defined to be − ]x′,x]dg(y)f(y). Let z− ∈A−f and z+ ∈A+f. We shall assume that z− <z+,
Note that ξφ is non-negative; this is closely related to the the argument being the same if the order is reversed. Define
R R
strictly positive gap condition. x− = max{A−∩[z−,z+]} and x+ = min{A+∩[x−,z+]}.
f f f f f
One of the main results in [1] is that for a non-decreasing It follows that ]x−,x+[∩A = ∅. Since A = A+ ∪A− it
fixed-point solution we have f f f f f f
follows from the stricly positive gap conditon (16) that x+−
f
ξ (w;f,g;x ,x )=φ(h ,h ;g(x +),f(x +)). (16) x− <2W. Define z =(x++x−)/2.
φ 1 2 f g 1 2 f f f
Setting x = x = z, it follows from the form (17), the
1 2
Note that if x = x = x then the right hand side is (the
1 2 strictly positive gap condition, and (16) that the df(x)dg(y)-
rightcontinuousversionof)I (x)−f(x)g(x).Bythestrictly
f,g measure of at least one of the following
positive gap condition we now have ξ (w;f,g;x ,x ) > 0
φ 1 2
for all x1,x2 where f(x1),g(x2) ∈]0,1[. Let the support of T1 ={(x,y):x≥z,y ≤z,x−y <W}
w be [−W,W] (we may have W = +∞). We define Ax = T2 ={(x,y):x≤z,y ≥z,y−x<W}
(x−W,x+W). From (15) it is easy to see that dg(A )=0
x
is strictly positive. Let us assume dfdg(T ) > 0. Note that
implies ξ (w;f,g;x,x) = 0 so dg(A ) > 0 for all x with 1
φ x x+ = minA+ ∩ [z,z + W] so it follows that there exists
g(x)∈]0,1[ and, similarly df(Ax)>0 for all x with f(x)∈ f f
]0,1[. These conditions imply f˙w(x) > 0 and g˙w(x) > 0 xg ∈]z−W,z]suchthat0≤x+f −xg <W. Thisclearlygives
for f(x) ∈]0,1[ and g(x) ∈]0,1[ respectively, where the dot x−f −xg < W. In addition we have x−f −xg > x−f −z >
denotes differentiation. −W so we obtain |x−f −xg| < W. The argument assuming
We express ξφ in a more useful form for our current dfdg(T2)>0 is similar.
purpose. We assume that x = x since we need the result Acknowledgments. R. E. thanks Vahid Aref for many inter-
1 2
only for this case. We claim that (15) is equal to esting discussions.
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