IndividualDistortionCriteria
JayanthNayak∗,ErtemTuncel∗,DenizGunduz†,ElzaErkip†
ofCalifornia,Riverside,CA
Email:{jnayak,ertem}@ee.ucr.edu†PolytechnicUniversity,Brooklyn,NY
E-mail:dgundu01@utopia.poly.edu,elza@poly.edu
Abstract—Thewell-knownsuccessiverefinementscenarioisextendedtovectorsourceswhereindividualdistortionconstraintsareposedoneveryvectorcomponent.Thisextensionisthenutilizedforthederivationofanecessaryandsufficientconditionforvectorsuccessiverefinability.For2-DvectorGaussianandbinarysymmetricsources,itturnsoutthatsuccessiverefinabilityisnotgrantedeverywhere,unlikeinthe1-Dcaseforthesamesources.Moreover,thebehaviorofthesesourceswithrespecttosuccessiverefinabilityexhibitremarkablesimilarity.
∗University
i=1,2,...,N,wehave
R(D)=
ˆ)}≤DE{d(X,X
min
ˆ)I(X;X
I.INTRODUCTION
Weextendthewell-knownsuccessiverefinementscenario
tovectorsourceswhereindividualdistortionconstraintsareposedoneveryvectorcomponent.Thesingle-layercounterpartofthisproblemwasaddressedin[7],anditwaslaterarguedin[4]thatthesingle-layerversionisinfactaspecialcaseofwhatisreferredtoasrobustdescriptions[1].
Inthiswork,theachievabilityregionofthescalarsuccessiverefinementproblem,whichwasderivedindependentlybyKoshelev[3]andRimoldi[5],isextendedtocoverthevectorsourcesunderindividualdistortioncriteriainastraightforwardmanner.Thisextension,inturn,isutilizedforthederivationofanecessaryandsufficientconditionfor“vectorsuccessiverefinability.”Notsurprisingly,thisconditionisalsoastraight-forwardextensionoftheMarkovityconditionderivedin[2].WethenusetheMarkovityconditiontoinvestigatewhethervectorsuccessiverefinabilityholdsfortwointerestingcases:(i)2-DGaussianvectorsandsquare-errorcriteriononeachvectorcomponent,and(ii)2-DbinarysymmetricvectorsandHammingdistortiononeachcomponent.Unlikeinthescalarcase,successiverefinabilityisnotgrantedeverywhere(i.e.,fromanydistortionvectorinthefirststagetoanydistortionvectorinthesecondstage)forthesetwoexamples.Also,thebehaviorofthesesourceswithrespecttosuccessiverefinabilityexhibitremarkablesimilarity.
II.EXTENSIONOFTHEACHIEVABILITYREGIONFOR
VECTORSOURCESTOMULTI-STAGECODINGWefirstrepeatherethesingle-lettercharacterizationin[7]forsingle-stagecoding.DenotingthememorylesssourceandtheauxiliaryreconstructionvariablebyX=
ˆ2...Xˆ=[Xˆ1XˆN]T,respectively,[X1X2...XN]TandX
ˆ)fordi(Xi,Xˆi)withandusingtheshorthandnotationd(X,X
whereweusetheconventionthata≤bmeansai≤bifor
i=1,2,...,N.Observethatthisresultisaspecialcaseoftherobustdescriptionsresultin[1].Morespecifically,intherobustdescriptionsscenario,individualdistortioncriteriaare
ˆi)}≤Di.allowedtobeoftheformE{di(X,X
ThefollowinglemmaextendsthisresulttoLstages.
Lemma1:AdistortionvectorsequenceD1≥D2≥···≥DLandacumulativeratesequenceR1≤R2≤···≤RLareachievable1ifandonlyifthereexistauxiliaryvectorsˆ1,Xˆ2,...,XˆLsatisfyingX
ˆ1,...,Xˆl)≤I(X;X
ˆl)}≤E{d(X,X
Rl
Dl
foralll=1,2,...,L.
Weomittheproof,sinceitisastraightforwardextensionoftheproofsin[3],[5].
Corollary1:The4-tuple{D1,D2,R(D1),R(D2)}is
ˆlachievingachievableifandonlyiftheoptimalvectorsX∗
{Dl,R(Dl)}forl=1,2satisfytheMarkovchain
ˆ2−Xˆ1.X−X∗∗
1ˆ2ˆProof:If(X,X)achieves{D1,D2,R(D1),R(D2)},
wehave
(a)(b)11ˆR(D)≥I(X;X)≥R(D1)and
ˆ1,Xˆ2)≥I(X;Xˆ2)≥R(D2)R(D2)≥I(X;X
where(a)and(c)followfromLemma1,(d)fromthechain
rule,and(b)and(e)fromthedefinitionofR(D).Thus,we
ˆ1=Xˆ1,Xˆ2=Xˆ2,andX−Xˆ2−Xˆ1.musthaveX∗∗∗∗
(c)
(d)
(e)
1-4244-1564-0/07/$25.00 ©2007 IEEE319
individualHammingdistortion.ItturnsoutthatforcertainvaluesofD1andD2,thesesourcesarenotsuccessivelyrefinable.
A.2-DGaussianSources
LetthecovariancematrixofthesourceXbegivenby
1ρ
CX=
ρ1where0<ρ<1.Forsimplicity,wewillusethenotation
jj=1−Diwithproperi,j.Definethreeδi=1−Diandδi
regionsintheunitsquareontheD-planeas
D1D2
==
{D:ρ≤δ1δ2}
δ1δc
D1∩D:ρ2≤min,
δδ
2
10.90.80.70.6ρ2D3D1δ20.50.40.30.20.10δ1δ2=ρ2D2D300.10.20.30.40.50.60.70.80.91ρ2δ1Fig.1.
ThethreeregionsD1,D2,andD3forρ=0.4.
Casei:D
D
δ2δ2−
δδ
δδ
δδ
δ1δ1
δ2δ2−
δ1δ1
δδ
δˆXδ
δ
2
≤
,o
δ1
=
δ2
.
320
10.90.80.70.6ρ2
10.90.80.70.6ρ2δ20.50.40.30.20.10δ2√ρ√,ρνν0.50.40.3√ρ√ν,ρνν=0.8ρ2
0.20.10ν=0.8ρ200.10.20.30.4δ1
0.50.60.70.80.9100.10.20.30.4δ10.50.60.70.80.91(a)10.90.80.70.6(b)10.90.80.70.6ρ2ρ2δ20.50.40.30.20.10√ρ√,ρννδ20.50.40.3ν=0.8ρ20.20.10ν=0.8ρ200.10.20.30.4δ10.50.60.70.80.9100.10.20.30.4δ10.50.60.70.80.91(c)(d)
2,δ2)-planeforseveral(δ1,δ1)pairs(indicatedusing◦)satisfyingδ1=νδ1andν=0.8.WesetFig.2.Thesuccessiverefinabilityregioninthe(δ121221
ρ1=ρ=0.4.Theparticularchoicesofδ1for(a)-(d)are0.05,0.3,√ν
Dνδ.
δ1δ1νδ1
νδ1
√ν
√,ρν
1
ρ2
δ1δ1νδ1
anνsatisfies(3)withequality,and
theaboveclaimiscorroborated.However,itisalsoclearfrom(3)thatthesuccessiverefinabilityregionisnotlimitedtothatrectangularregion.
Figure2showsthesuccessiverefinabilityregionforseveral
11
choicesof(δ1,δ2).
√
ν
B.2-DBinarySymmetricSources
Letthepmfofthesourcebegivenby
1
−p2PX=
−2
4
aν.T
ν)i
≤
2.
I
4,
t
fo
2
√,ρ
ν
ν).I
321
Wefirstcomputetherate-distortionfunctionandtheoptimaltestchannelsforthesingle-stageproblem.
Theorem1:Therate-distortionfunctionfor2-Dbinarysymmetricsourcesisgivenby
⎧
H(X)−H(D1)−H(D⎪2)⎪D∈E1⎪⎪⎨H(X)−H(2p)−2pHD1+D2+2p−1
4pR(D)=
⎪⎪2(1−2p)⎪⎪⎩
D
,δ
δ
δ2
δ2
δ2δ2
δ
⎦
δ2
δ2
2
−
2
−
4
⎣
δ2δ2
⎦
4
1
δδ
D
2
2
δ2
δ2
2
4
⎣
δ2
δ2
+
δ2
δ2δ2
−
δ2
δ2
−−
⎦
δ2
δ2
=
δ2
≤
δ
,o
ν
D
4p−1
an
.I
δ2
δ2
δ2δ2
⎦
4δ2δ2
322
10.90.80.70.64p−110.90.80.70.64p−1δ20.50.40.30.20.10δ24p−1ν(4p−1)ν,0.50.40.34p−1ν,ν(4p−1)ν=0.84p−1
0.20.10ν=0.84p−100.10.20.30.4δ1
0.50.60.70.80.9100.10.20.30.4δ10.50.60.70.80.91(a)10.90.80.70.6(b)10.90.80.70.64p−14p−1δ20.50.40.30.20.10δ24p−1ν,ν(4p−1)0.50.40.3ν=0.84p−10.20.10ν=0.84p−100.10.20.30.4δ10.50.60.70.80.9100.10.20.30.4δ10.50.60.70.80.91(c)(d)
2,δ2)-planeforseveral(δ1,δ1)pairs(indicatedusing◦)satisfyingδ1=νδ1andν=0.8.ToFig.3.Thesuccessiverefinabilityregioninthe(δ121221
2withρ=0.4.Theparticularchoicesofδ1for(a)-(d)are0.05,0.3,emphasizethesimilaritytotheGuassiancase,wesetp=0.29sothat4p−1=ρ1q4p−1
ν
=
ν
an
ν(4p−1).I
ν
,ν(4p−1)
p(ˆx)e−β(x⊕xˆ)e−β(x⊕xˆ)
≤
323
ˆ.Thecorrespondingbackwardchannelischaracterizedforallxby
ˆ)=pX|Xˆ(x|x
ˆ1)−β2(x2⊕xˆ2)
pX(x)e−β1(x1⊕xe
.ˆ)e−β(x⊕xˆ)p(ˆx)e−β(x⊕x
2
2.I
−
2
−
1+sts+t
s+t1+st
(s
(1−s)(1−t)
2
2
1+sts+t
4p2(1−2p)
(1+s)(1+t)
≤
(1+s)(1+t)
sx
2(1−2p)
4p
1+s1+s
1+t1+t
1−D1−D
.
2
at
2
2
(1+s)(1+t)
≥
s+t1+st
324
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