Sparse Ternary Codes for similarity search have higher coding gain than dense binary codes

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by  Sohrab Ferdowsi

SPARSETERNARYCODESFORSIMILARITYSEARCH
HAVEHIGHERCODINGGAINTHANDENSEBINARYCODES
SohrabFerdowsi,SlavaVoloshynovskiy,DimcheKostadinov,TarasHolotyak
DepartmentofComputerScience,UniversityofGeneva,Switzerland
sohrab.ferdowsi,svolos,dimche.kostadinov,taras.holotyak @unige.ch
{ }
ABSTRACT on Nearest Neighbors (NN’s) to a given query within the
7
1 This paper addresses the problem of Approximate Nearest items in a database. This problem is the basis for many
0 applications like similar image retrieval, copy detection,
Neighbor(ANN)searchinpatternrecognitionwherefeature
2 copy-write protection, etc. The idea is that semantic simi-
vectorsinadatabaseareencodedascompactcodesinorderto

n larity can be mapped to distances within a vectorial space
speed-upthesimilaritysearchinlarge-scaledatabases. Con-
a Rn. The search for similarity then reduces to search for
sideringtheANNproblemfromaninformation-theoreticper-
J
NN’s by comparing the vectorial representation of a given
spective,weinterpretitasanencodingwhichmapstheorig-
6
query to representations of items in a database. The chal-
2 inal feature vectors to a less-entropic sparse representation
lenge, however, is when M is huge, e.g., several billions.
whilerequiringthemtobeasinformativeaspossible.Wethen
] define the coding gain for ANN search using information- TheNNsearchbasedonlinearscanthenbecomesthemain
T bottleneck for the system. Approximative solutions where
theoreticmeasures. Wenextshowthattheclassicalapproach
I performanceistradedwithmemoryandcomplexityarethen
. tothisproblemwhichconsistsofbinarizationoftheprojected
s to be preferred. This is addressed in Approximate Nearest
c vectorsissub-optimal. Instead,weshowthatarecentlypro-
[ posedternaryencodingachieveshighercodinggains. Neighbor (ANN) search, an active topic in computer vision
andmachinelearningcommunities.1
1 Index Terms— Approximate Nearest Neighbor search,
The methods introduced for ANN in pattern recognition
v
content identification, binary hashing, coding gain, sparse
5 communitiescanroughlybedividedintotwomaincategories.
representation
7 A first category of methods is based on quantization of the
6 data.Thesemethodsmainlytargetmemoryconstraintswhere
7
1. INTRODUCTION every database item is given an index, or a set of indices
0
whichrefertocodeword(s)from atrainedcodebook. Atthe
.
1
The problem of content identification, e.g., identification of query time, the items are reconstructed from the codebooks
0
peoplefromtheirbiometricsorobjectsfromunclonablefea- and matched with the given query, potentially through look-
7
1 tures was first formalized in information-theoretic terms by up-tables.
: Willemset.al.in[1].Theydefinedtheidentificationcapacity AsecondfamilyofmethodsforANN,alsoconsideredin
v
i astheexponentofthenumberofdatabaseitemsM thatcould thiswork,isbasedonprojectingthedatafromRntoausually
X
reliablybeidentifiedinanasymptoticcasewherethefeature lowerdimensionalspace. Theprojecteddataarefurtherbina-
r dimensionn . Theymodeledtheenrollmentandacqui- rizedusingthesignfunction. Thesearchinthebinaryspace
a sition system→s as∞noisy communication channels while they isfastersinceitislessentropicthantheoriginalRn. More-
consideredthevectorsasrandomchannelcodes. Theauthors over,thisapproachbringsmorepracticaladvantagesincethe
thencharacterizedtheidentificationcapacityasI(F;Q),the distancematchinginthebinaryspaceusingtheHammingdis-
mutualinformationbetweentheenrolleditemsandthenoisy tanceisperformedinfixed-pointinsteadofthefloatingpoint
queries. Inthissetup, however, theincreaseinnleadstoan operationsintheoriginalrealspaceorthespaceofcodebooks
exponentialincreaseinM 2nI(F;Q). Thisincursinfeasible inquantization-basedmethods.
(cid:39)
searchandmemorycomplexities,makingthesystemimprac- While the content identification literature is rigorously
tical. Subsequent works attempted at decreasing these com- studying the fundamental limits of identification under stor-
plexities. Forexample,[2]consideredatwo-stageclustering- ageorcomplexityconstraintsforknownsourceandchannel
basedsystemtospeed-upthesearch,while[3]consideredthe distributionsandinfinitecode-lengths,theliteratureofANN
compression of vectors prior to enrollment and studied the search drops these information-theoretic assumptions and
achievablestorageandidentificationrates. hence treats the problem from a more practical perspective.
Similar to the content identification problem, the pattern
recognitioncommunityconsidersthesimilaritysearchbased 1Referto[4]fordetailedreviewofANNmethodsandapplications.
This work tries to bridge these approaches. While we do In the case of the content identification problem, the
not assume the asymptotic case and hence do not consider search system finds the nearest item from to q, i.e.,
fundamental limits with their achievability and converse ar- ˆi = argmin[ (q,f(i))],orproducesalist F(q)ofnearest
guments,weproposeapracticalsystemswhereitisexplicitly 1(cid:54)i(cid:54)M D(cid:96)2 L
items in the more general (A)NN search, i.e., (q) = i :
required to maximize the mutual information between the (q,f(i))(cid:54)(cid:15)n ,where(cid:15)>0isadefinedthrLeshold. {
query and database codes. Moreover, to address complexity D(cid:96)2 }
requirements, we further require the codes to be sparse and
henceallowingefficientsearchandstorage. Concretely, this 2.2. Codinggainforsimilaritysearch
workbringsthefollowingcontributions:
Theaboveproblemformulationimpliesamemorylessobser-
Considering the projection approach to ANN, we intro- vation channel for queries. In [1] the identification capacity
•
duce the concept of “coding gain” to quantify the effi- isdefinedforthischannel(alongwithanotherAWGNforthe
ciencyofacodingschemeforsimilaritysearch. enrollmentchannel)asameasureforthenumberofitemsthat
Wethenshowthatthestandardapproachintheliterature canbereliablytransmitted.Whilethisanalysisisbasedonthe
• which is based on binarization of the projected values is asymptoticassumptionwherethenumberofitemsM should
sub-optimalascharacterizedbythecodinggain. Instead, growexponentiallywithdimensionn,itcanbeargued,how-
we show that the ternarization approach proposed in [5], ever, that M is fixed in practice and can be even well below
namely the Sparse Ternary Codes (STC) achieves more theamountthatthecapacitywouldaccommodate. Moreover,
favorable gains while keeping all the advantages of the theprobabilityofcorrectidentification,mightbeonlyasymp-
project-and-quantizeapproach. toticallyachievableandislessthan1inpractice.
Therefore,inthiswork,insteadofthechannelcapacityar-
WenextshowhowtodesigntheSTCcodestomaximize
• guments,weconsiderthepracticalcasewherethefocusison
the coding gain. As opposed to the Maximum Likeli-
fast decoding for which the given database is encoded. This
hood (ML) decoder, we consider a sub-optimal but fast
comeswiththepriceofloweridentificationperformance.
decoderwhichbringssub-linearsearchcomplexity. This
To quantify the efficiency of a coding scheme for fast
provides a wealthof possibilities for design, considering
ANN search under the setup of section 2.1, we consider the
thememory-complexitytrade-offs, whichismuchricher
coding gain as the ratio of mutual information between the
than the binary counterpart. Based on this decoder, we
encodedversionsoftheenrolleditemsandquery,i.e.:
simulateanidentificationscenariotoshowtheadvantages
oftheproposedmethodology. I(X;Y)
g (ψ)= , (1)
Theproblemformulation,alongwiththedefinitionofthe F H(X)
proposedcodinggainarediscussedinsection2.Binarycodes wheretheencodingforenrollmentprovidesX = ψ [F]and
e
aresummarizedinsection3whiletheternaryencodingisdis- encodingforidentificationprovidesY =ψ [Q]:2
i
cussedinsection4.Comparisonofthetwoencodingschemes
densebinarycodes
in terms of the coding gain and also complexity ratio is per-
+1 +1
formedinsection5. Finally,section6concludesthepaper. X Y
F ψe X 1 1
obs. ch. − −
SparseTernaryCodes
2. PRELIMINARIES P + p(y|x) +1 +1
X 0 0 Y
2.1. Problemformulation Q ψi Y 1 1
− −
Fig. 1: Encoders ψ () and ψ () map the feature space to
e i
Weconsideradatabase = f(1), ,f(M) ofM items the space of codes for·enrollmen·t and identification, respec-
F { ··· }
asvectorialdata-pointsf(i)(cid:48)swheref(i)=[f1(i), ,fn(i)]T tively. ABSCmodelstheperturbationinbinarycodes,while
Rn and1 (cid:54) i (cid:54) M. Weassume,forthesakeo·f··analysis, anoise-adaptiveternarychannelmodelstheSTC.
∈
that each database entry f(i) is a realization of a random
vectorFwhosenelementsarei.i.d. realizationsofarandom MutualinformationinthedefinitionofEq. 1candirectly
variableF withF (0,σ2). belinkedtochanneltransitionprobabilities.Thechoiceofen-
∼N F
The query q = [q , ,q ]T Rn is a perturbed ver- tropy,ontheotherhand,isjustifiedbybothmemoryandcom-
1 n
··· ∈
sion of an item from . We assume the perturbation fol- plexityrequirements.Obviously,thecostofthedatabasestor-
F
lows an AWGN model as Q = F+P, where elements of age is directly linked to H(X) when source coding is used.
P (0,σ2I )areindependentfromelementsofF. Fora Moreover,sincetheeffectivespacesizeis n 2nH(X),a
∼N P n |X |≈
givenquery,thesimilaritysearchisbasedonadistancemea- lowerentropicspacealsoimpliesalowersearchcomplexity.
sure (, ) : Rn Rn R+,whichisassumedthroughout
D · · × → 2Noticethat,unliketheusualbinarydesign,intheSTCframework,these
the paper to be the Euclidean distance, i.e., (q,f(i)) =
D(cid:96)2 twostagesneednotbethesame. Infact,theyaredesignedaccordingtothe
q f(i) 2. statisticsoftheobservationchannel.
|| − ||
3. BINARYCODES First,thecostofstorageofanydatabasevectorwillreduceto
l bits. Second,thesearchisperformedonthebinaryvectors
Binaryencodingistheclassicalapproachwidelyusedinthe usingfixed-pointoperations.However,aswewillshowinthe
pattern recognition literature to address fast search methods. next chapter, better performance can be achieved with STC,
Asmentionedearlier,theyarebasedonprojectingthedatato theternarycounterpartofthebinarycodes.
alower-dimensionalspaceandbinarizingthem.
4. SPARSETERNARYCODES(STC)
3.1. RandomProjections
Theprojectiononvectorsof andqinRn isperformedus- TheideabehindtheSTC[5]isthatdifferentprojectionvalues
ing random projections to RFl.3 This choice is justified by havedifferentrobustnesstonoisewhichcanbeexpressedas
abitreliability. Thiscanbeachievedbyignoringthevalues
theoremslikeJohnson-Lindenstrausslemma[6]where,under
whose magnitude is below a threshold, i.e., X = φ (F˜)
certainconditions,thepair-wisedistancesintheprojecteddo- t λX
andY =φ (Q˜),where
mainareessentiallypreservedwithprobabilisticguarantees. t λY
Randomprojectionsareusuallyperformedusingann l 
unit-normmatrix whoseelementsaregeneratedi.i.d.fr×om +1 t(cid:62)λ,
a Gaussian distribWution W (0, 1). In the case of bi- φλ(t)= 0 λ<t<λ,
nary codes, for memory con∼straNints, unsually l < n. For the  1 −t(cid:54) λ,
− −
ternary codes proposed in [5] which we analyze next, how-
ever,sincetheentropycanbeboundedbyimposingsparsity, and λX and λY are threshold values for the enrolled items
longer code lengths can be considered. In this case, the use andthequery,respectively. ThethresholdvaluesλX andλY
ofsparserandomprojectionsof[7]isjustifiedwhereperfor- directlyinfluencethesparsityratiosα,theratioofthenumber
manceguaranteesrequirelongerlengths. Thisisveryuseful ofelementsofXt withnon-zerovalues( 1 ’s)tothecode
{± }
since the sparsity in the projector matrix can be reduced to lengthlandγ,theratioofthenumberofelementsofYtwith
(2nl)insteadof (nl),wheresisthesparsityparameteras non-zerovaluestothecodelength.Thesevaluescanbeeasily
O s (cid:40) (cid:112) s Ow.p. 2, calculatedas:
inW ± 2n s
∼ 0 w.p. 1− 2s. α=(cid:90) −λX p(f˜)df˜+(cid:90) ∞p(f˜)df˜=2 (cid:16)λX(cid:17),
According to the data model assumed in section 2.1, the
Q σ
elementsoftheprojecteddataF˜andQ˜willalsofollowGaus- −∞ λX F
sian distribution as F˜ (0,σ2) and Q˜ F˜ (F˜,σ2), γ =(cid:90) −λY p(q˜)dq˜+(cid:90) ∞p(q˜)dq˜=2 (cid:16) λY (cid:17).
whereF˜T = FTW an∼dQ˜NT = QFTW. Th|isg∼iveNsthejoiPnt- −∞ λY Q (cid:112)σF2 +σP2
distributionoftheprojecteddataasabivariateGaussianwith (2)
(cid:20) (cid:21)
σ2 σ2
ρ= √σσ2F+σ2 ,i.e.,p(f˜,q˜)=N([0,0]T, σF2 σ2 +Fσ2 ). While in [5], a signal approximation-based justification
F P F F P
for the choice of φ was proposed, here we propose an
λ
information-theoreticargument.
3.2. Binarizingtheprojections
Binary codes are provided by binarizing the projected val-
4.1. InformationmeasuresforSTC
ues using the sign function, i.e., X = sign(F˜) and Y =
b b
sign(Q˜). One can consider an equivalent to the observation It is very useful to study the equivalent ternary channel be-
channel in the encoded case. In [8], this channel was de- tween Xt and Yt. The element-wise entropy of the ternary
rived as a BSC with I(X ,Y ) = 1 H (P ), where P = codeis:
b b 2 b b
−
Ep(q˜)[Q(σ|q˜P|)]= π1 arccos(ρ)istheprobabilityofbitflipand H(X )= 2αlog (α) (1 2α)log (1 2α). (3)
Q(u)=(cid:82)u∞ √12πe−2u(cid:48)du(cid:48)istheQ-function. t − 2 − − 2 −
The maximum-likelihood optimal decoder for the bi- AccordingtoEq.2,thisisonlyafunctionofthethreshold
nary codes is simply the minimum Hamming-distance de- valueλ (seeFig. 2.a).
X
coder, i.e., ˆi = argmin[ (y ,x (i))] with (y,x) = As for I(X ;Y ), unlike the binary case we cannot have
H b b H t t
1(cid:54)i(cid:54)M D D aclosedformexpression. Instead,wecalculatethisquantity
1l (cid:80)lj=1yj ⊕xj and ⊕ representing the XOR operator. As withnumericalintegration.
mentioned earlier, this has two main practical advantages. Considerthetransitionprobabilitiesforternarychannel:
3Whilerecenttrendinpatternrecognitionisfavoringprojectorswhichare  
p p p
learnedfromthedata,theperformanceboostobtainedfromthesemethodsis +1+1 0+1 1+1
limitedtolowbit-rateregimesonly. Moreover,theyarenotstraightforward P =p(y x)= p+1| 0 p0| 0 p− 1| 0 . (4)
foranalyticpurposes. | p | p | p − |
+1 1 0 1 1 1
|− |− − |−
The elements of the transition matrix P are defined as inte-
grationoftheconditionaldistributionp(Q˜|F˜)= p(pF(˜F˜,Q˜)) with (cid:88)n (cid:104)
proper integral limits and calculated numerically for these ˆi=argmax ν(1 +1 )
t(cid:82)hλ∞rYes(cid:82)(cid:82)hλλ∞∞oXXldpp((fvf˜˜)a,dq˜l)fu˜def˜sd.q˜Fiosrthexeapmropblea,bpil+it1y|+th1a=t vaplQu˜|eF˜s(o+f1F˜|+gr1e)at=er 1(cid:54)i(cid:54)+Mν(cid:48)j(=11{xj=+1{,xyjj==y−j1=}++1}1{xj={x−j1=,yyjj==−+11}})(cid:105), (7)
than λX are queried with values greater than λY while where ν and ν(cid:48) are voting constants that will be specified
p0|+1 = pQ˜|F˜(0| + 1) = (cid:82)−λλYY(cid:82)(cid:82)λ∞λX∞Xpp(f(˜f)˜d,q˜f˜)df˜dq˜ quantifies shorTtlhyisanddec1odiserthisesiunbd-iocapttoimrfaulnincttihoen.MLsensesinceinstead
the probability that these values have magnitudes less than
oftakingintoaccountallthe9transitions(fromwhich5in-
λY in the query and hence will be assigned as ‘0’ in the dependent),itisconsideringonly4transitions(fromwhich3
ternaryrepresentation. Outofthese9transitionprobabilities,
independent)andtreatingtheother5asthesame. However,
5areindependentandtherestarereplicatedduetosymmetry.
itisperformingthesearchinsub-linearcomplexityinsteadof
Themutualinformationintheternarycaseis: theotherwiseexhaustivescan. Aswewillshownext,thisal-
lowsustodrasticallyreducethesearchcomplexity. Analgo-
I(Xt;Yt)=H(Xt)+H(Yt) H(Xt,Yt), rithmicdescriptionofthisdecoderwasgivenin[5]forpracti-
−
calscenarios. Inthiswork,however,sinceweareassuminga
where H(Xt) is given by Eq. 3. Similarly, H(Yt) = knownsetupanddistributionforthedata,wecancalculatethe
−2γlog2(γ) − (1 − 2γ)log2(1 − 2γ), where γ can be de- optimalvaluesofν andν(cid:48) asEq. 8,whereν encouragessign
rived from its definition of Eq. 2 or, equivalently γ = matchandν isanegativevaluethatpenalizessignmismatch.
(cid:48)
α(P(1,1)+P(1,3))+(1 2α)P(1,2). Thejointentropy Sinceweareignoringthe‘0’transitions,ν0 compensatesby
−
isthencalculatedfromelementsofP as: considering the expectation of the occurrences of logp(y x)
|
termsinEq. 6,wheneitherxoryis‘0’.
H(X ,Y )= 2αP(1,1)log(αP(1,1))
t t − ν =ν0+log(P(1,1)),
2αP(1,2)log(αP(1,2)) 2αP(1,3)log(αP(1,3))
− − ν =ν0+log(P(1,3)),
(cid:48)
2(1 2α)P(2,1)log((1 2α)P(2,1))
− − − ν0 = (cid:2)2αP(1,2)logP(1,2)
(1 2α)P(2,2)log((1 2α)P(2,2)). −
− − − (cid:3)
(5) +(1 2α)(2P(2,1)logP(2,1)+P(2,2)logP(2,2)) .
−
(8)
This, in fact, is a function only of λ , λ and σ2. We will
X Y P
usethesequantitiesinsection5.1tocomparethecodinggain 5. RESULTS
oftheseternarycodeswiththoseofbinarycodes.
5.1. ComparisonofSparseTernaryCodeswithdensebi-
narycodes
4.2. Fastsub-lineardecoder
HerewecomparethecodinggainofSTCwithdensebinary
Given a ternary sequence y = (y , ,y )T, the codes. Since the mutual information of the ternary code de-
1 l
maximum-likelihood rule to choose the··o·ptimal x(i) pends on both λ and λ , for different values of λ 4, we
X Y X
amongst the registered x(1), ,x(M) is given as ˆi = chooseλ thatmaximizesI(X ;Y ). Sincethereisnoana-
Y t t
···
palrigfimedaxas1(cid:54)thi(cid:54)eMmapxXim|Yi(zxat(iio)n|Yoft=heylo)g.-lTikheilsihrouoledacsa:n be sim- lλytic,awleeuxspereasssiimonplfeorgrtihdesmeaarxcihmaummonogfdIi(fXfetr;eYntt)v,aflouresaofifxλed
X Y
forthemaximizerofmutualinformation. Fig. 2.dshowsthe
l (λ ,λ ) pairs under three different noise levels, as charac-
(cid:88) X ∗Y
ˆi=a1r(cid:54)gim(cid:54)Max logpY|X(yj|xj). (6) terizedbySNR=10log10σσF22 .
j=1 P
Forfaircomparison,wechoosethecodelengthsofbinary
andternarycases(l andl ,respectively)toensurethesame
Thisrequirestotakeintoaccountallthe9transitionprobabil- b t
code entropy, i.e., l H(X ) = l H(X ). We then compare
itiesofEq. 4. b b t t
l I(X ,Y )withl I(X ,Y )inFig. 2.b.
Similar to the binary codes, the above maximum- b b b t t t
Asisseenfromthisfigure,theproperchoiceofthresholds
likelihooddecodershouldexhaustivelyscanalltheM items
leadstointerestingregimeswheretheternarycodepreserves
in the database. However, by skipping the transitions to and
moremutualinformationcomparedtobinarycodes.
from‘0’andonlyconsideringelementsofP containing‘+1’
and‘-1’,analternativedecodercanbeconsideredas: 4Inpractice,thisischosenbasedonmemoryconstraints.
Gaussian(SNR= 10dB) Gaussian(SNR=0dB) Gaussian(SNR=+10dB)
−
binary(SNR= 10dB) binary(SNR=0dB) binary(SNR=+10dB)
−
STC(SNR= 10dB) STC(SNR=0dB) STC(SNR=+10dB)
−
(a)entropy (b)mutualinformation (c)codinggain (d)optimalthresholdvaluesforSTC
100
100 100
3
10−1
10−1
10−2 λ∗Y2
h(F) 10−1
10−2 H(Xb) 10−3 1
H(Xt)
10−3 10−4
0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
λX λX λX λX
Fig.2:Codinggaincomparisonforternaryandbinaryencoding.AlthoughtheI(X ;Y )curvehasaconcaveshapewithrespect
t t
toλ whichmeansthatitsvaluedecreasesasthecodebecomessparser,H(X )decreaseswithafasterrateforincreasingλ .
X t X
ThismeansthatthecodinggainI(X ;Y )/H(X )increasesasthecodebecomesparser.
t t t
REFERENCES
binary(SNR= 10dB) binary(SNR=0dB) binary(SNR=+10dB)
−
STC(SNR= 10dB) STC(SNR=0dB) STC(SNR=+10dB)
−
(a)memoryusageperitem (b)complexityratio [1] F.Willems,T.Kalker,J.Goseling,andJ.P.Linnartz,“On
100 100 the capacity of a biometrical identification system,” in
IEEE International Symposium on Information Theory,
10−1 10−1 2003.Proceedings.,2003,pp.82–.
Pd Pd
[2] F.M.J.Willems,“Searchingmethodsforbiometriciden-
10−2 10−2
tification systems: Fundamental limits,” in 2009 IEEE
International Symposium on Information Theory, 2009,
101 102 103 10−5 10−4 10−3 10−2 10−1 100 101 pp.2241–2245.
memory(bits) complexityratio
[3] E. Tuncel, “Capacity/storage tradeoff in high-
Fig.3: Performance-memory-complexity profile for identifi-
dimensionalidentificationsystems,” IEEETransactions
cation of M = 1 Mio synthetic data with n = 500. The
on Information Theory, vol. 55, no. 5, pp. 2097–2106,
sub-lineardecoderofsection4.2wasusedforSTC.
2009.
[4] Jingdong Wang, Heng Tao Shen, Jingkuan Song, and
5.2. Identificationperformance Jianqiu Ji, “Hashing for similarity search: A survey,”
CoRR,vol.abs/1408.2927,2014.
We consider the identification of synthetic data by compar-
ingtheprobabilityofcorrectidentificationfordifferentvalue [5] S. Ferdowsi, S. Voloshynovskiy, D. Kostadinov, and
pairsofmemoryandcomplexityratioinFig. 3. Memoryus- T. Holotyak, “Fast content identification in high-
age is measured by entropy of a coded block, i.e., l H(X ) dimensional feature spaces using sparse ternary codes,”
b b
for the binary code and l H(X ) for ternary and complexity in 2016 IEEE International Workshop on Information
t t
ratio is measured as lb and 4αγlt for the binary and ternary ForensicsandSecurity(WIFS),2016,pp.1–6.
n n
codes, respectively. The projections are done using sparse [6] William B Johnson and Joram Lindenstrauss, “Exten-
randomprojectionsof[7],describedinsection3.1. Thecom- sions of lipschitz mappings into a hilbert space,” Con-
plexity of this stage is fixed for each of the experiments, for temporarymathematics.
bothbinaryandternarycases.
[7] PingLi,TrevorJHastie,andKennethWChurch, “Very
sparse random projections,” in Proc. of 12th ACM
6. CONCLUSIONS SIGKDD int. conf. on Knowledge discovery and data
mining.ACM,2006,pp.287–296.
ThisworkisanattempttoanalyzeSTC,arecentlyproposed
[8] S. Voloshynovskiy, O. Koval, F. Beekhof,
codingschemefortheproblemofcontentidentificationfrom
F. Farhadzadeh, and T. Holotyak, “Information-
an information-theoretic perspective. It was shown that the
theoretical analysis of private content identification,”
STCpossessesmorefavorablecodinggains, ascomparedto
in 2010 IEEE Information Theory Workshop, 2010, pp.
theclassicalbinaryhashing. Asafuturework,theextension
1–5.
tocompression-basedsetupsfortheSTCwillbeconsidered.