Table4: HandwritingSynthesis s Results. . Allresultsrecordedontheval-
idationset. ‘Log-Loss’ ’ is s the meanvalue of L(x) innats. . ‘SSE’is s themean
sum-squared-errorperdatapoint.
Regularisation
Log-Loss SSE
none
-1096.9
0.23
adaptiveweightnoise -1128.2
0.23
digits and d most t of the punctuation n characters replacedwitha generic c ‘non-
letter’label
2
.
Thenetworkarchitecturewasas similaraspossibletothe bestprediction
network: three e hidden n layers s of 400 0 LSTM M cells s each, 20bivariate e Gaussian
mixturecomponentsattheoutputlayerandasize3inputlayer. Thecharacter
sequencewasencodedwithone-hotvectors,andhencethewindowvectorswere
size57. Amixtureof10Gaussianfunctionswasusedforthewindowparameters,
requiringasize30parametervector. Thetotalnumberofweightswasincreased
toapproximately3.7M.
The network was trainedwith h rmsprop,using the e sameparameters s as s in
theprevioussection. The e networkwas retrainedwithadaptiveweight noise,
initialstandarddeviation0.075,andtheoutputandLSTMgradientswereagain
clippedintherange[ 100;100]and[ 10;10]respectively.
Table4showsthatadaptiveweightnoisegaveaconsiderableimprovement
inlog-loss(around31.3nats) butnosignicantchangeinsum-squarederror.
Theregularisednetworkappearstogenerateslightlymorerealisticsequences,
althoughthe dierence is s hardto discern by y eye. . Both h networks s performed
considerably betterthanthebest predictionnetwork. . Inparticularthesum-
squared-error wasreducedby44%. . This s is likelydueinlargepart totheim-
provedpredictionsattheendsofstrokes,wheretheerrorislargest.
5.3 UnbiasedSampling
Givenc,anunbiasedsamplecanbepickedfromPr(xjc)byiterativelydrawing
x
t+1
fromPr(x
t+1
jy
t
),justasforthepredictionnetwork. Theonlydierenceis
thatwemustalsodecidewhenthesynthesisnetworkhasnishedwritingthetext
andshouldstopmakinganyfuturedecisions. Todothis,weusethefollowing
heuristic: assoonas s (t;U+1)>(t;u)81uU thecurrentinputx
t
is
denedastheendof thesequenceandsamplingends. . Examples s of unbiased
synthesissamplesareshowninFig.15. Theseandallsubsequentgureswere
generated using the e synthesis network retrained d with adaptive weight noise.
Noticehowstylistictraits,suchascharactersize,slant,cursiveness etc. . vary
2
Thiswasanoversight;howeveritledtotheinterestingresultthatwhenthetextcontains
anon-letter,thenetworkmust select adigits or punctuation mark togenerate. . Sometimes
thecharactercanbebeinferred fromthecontext(e.g.theapostrophein\can’t");otherwise
it ischosenatrandom.
31
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widelybetweenthesamples,butremainmore-or-less consistent withinthem.
This suggests that thenetwork identies thetraits earlyoninthe sequence,
thenremembersthemuntiltheend. Bylookingthroughenoughsamplesfora
giventext,itappearstobepossibletondvirtuallyanycombinationofstylistic
traits,whichsuggeststhatthenetworkmodelsthemindependentlybothfrom
eachotherandfromthetext.
‘Blindtaste tests’carriedout by the author during presentations suggest
thatatleast some unbiasedsamples cannot bedistinguishedfromrealhand-
writing bythe humaneye. . Nonetheless s the network does s make e mistakes we
wouldnot expect a a humanwriter r tomake,ofteninvolvingmissing,confused
or garbledletters
3
; this s suggests that the network sometimes has trouble de-
terminingthealignmentbetweenthecharactersandthetrace. Thenumberof
mistakesincreasesmarkedlywhenless commonwordsorphrasesareincluded
inthe character sequence. . Presumablythisis s becausethenetwork learns an
implicitcharacter-levellanguagemodelfromthetrainingsetthatgetsconfused
whenrareorunknowntransitionsoccur.
5.4 BiasedSampling
One problem withunbiasedsamples is that they tendtobe dicult toread
(partly because real handwritingis dicult to read, and d partly y because the
network isanimperfect model). . Intuitively,wewouldexpect t the networkto
give higher probability togoodhandwritingbecause it tends tobe smoother
andmorepredictablethanbadhandwriting. Ifthisistrue,weshouldaimto
outputmoreprobableelementsofPr(xjc)ifwewantthesamplestobeeasierto
read. Aprincipledsearchforhighprobabilitysamplescouldleadtoadicult
inferenceproblem,astheprobabilityof every outputdependsonallprevious
outputs.Howeverasimpleheuristic,wherethesamplerisbiasedtowardsmore
probablepredictions ateachstepindependently,generally gives goodresults.
Dene the probabilitybias s bas s a a real number greater r thanor equaltozero.
Beforedrawingasample fromPr(x
t+1
jy
t
), eachstandarddeviation
j
t
inthe
GaussianmixtureisrecalculatedfromEq.(21)to
j
t
=exp
^
j
t
b
(61)
andeachmixtureweightisrecalculatedfromEq.(19)to
j
t
=
exp
^
j
t
(1+b)
P
M
j0=1
exp
^
j0
t
(1+b)
(62)
Thisarticiallyreducesthevarianceinboththechoiceofcomponentfromthe
mixture,andinthedistributionofthecomponentitself. Whenb=0unbiased
samplingisrecovered,andas b!1 thevarianceinthesamplingdisappears
3Weexpecthumanstomakemistakeslikemisspelling‘temperament’as‘temperement’,as
thesecondwriterinFig.15seemstohavedone.
32
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Figure15: Realandgeneratedhandwriting. Thetoplineineachblockis
real,therestareunbiasedsamplesfromthesynthesisnetwork. Thetwotexts
arefromthevalidationsetandwerenotseenduringtraining.
33
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andthenetworkalwaysoutputsthemodeofthemostprobablecomponentin
themixture(whichisnot necessarilythemodeof themixture,but at least a
reasonableapproximation). Fig.16showstheeectofprogressivelyincreasing
thebias,andFig.17showssamplesgeneratedwithalowbiasforthesametexts
asFig.15.
5.5 PrimedSampling
Another reasonto o constrain the sampling wouldbe to generate handwriting
inthestyleofa particular r writer(rather thanina a randomlyselectedstyle).
The easiest way to do this s wouldbe e to retrainit on n that t writer only. . But
evenwithoutretraining,itispossibletomimicaparticularstyleby‘priming’
thenetwork witharealsequence, thengeneratinganextensionwiththe real
sequencestillinthenetwork’smemory.Thiscanbeachievedforarealx,cand
asynthesischaracterstringsbysettingthe charactersequencetoc
0
=c+s
and clamping the e data inputs s to x x for the rst T T timesteps, , then n sampling
as usualuntil l the sequenceends. . Examples s of primedsamples are shownin
Figs.18and19. Thefactthatprimingworksprovesthatthenetworkisableto
rememberstylisticfeaturesidentiedearlieroninthesequence.Thistechnique
appearstoworkbetterforsequencesinthetrainingdatathanthosethenetwork
hasneverseen.
Primedsamplingandreducedvariancesamplingcanalsobecombined. As
showninFigs.20and21thistendstoproducesamplesina‘cleanedup’version
oftheprimingstyle,withoverallstylistictraits suchas slant andcursiveness
retained, but the strokes s appearingsmoother and d more e regular. . A A possible
applicationwouldbethearticialenhancementofpoorhandwriting.
6 ConclusionsandFutureWork
This paper has demonstratedtheability ofLongShort-Term Memoryrecur-
rent neuralnetworkstogeneratebothdiscreteandreal-valuedsequenceswith
complex,long-rangestructureusingnext-stepprediction.Ithasalsointroduced
anovelconvolutionalmechanismthatallowsarecurrent networktocondition
itspredictionsonanauxiliaryannotationsequence,andusedthisapproachto
synthesisediverseandrealisticsamplesofonlinehandwriting. Furthermore,it
hasshownhowthesesamplescanbebiasedtowardsgreaterlegibility,andhow
theycanbemodelledonthestyleofaparticularwriter.
Severaldirectionsfor futurework suggestthemselves. . Oneis s theapplica-
tionofthenetworktospeechsynthesis,whichis likelytobemorechallenging
thanhandwritingsynthesisduetothegreaterdimensionalityofthedatapoints.
Anotheristogainabetterinsightintotheinternalrepresentationofthedata,
andtousethisto manipulatethe sampledistributiondirectly. . Itwouldalso
beinterestingtodevelopamechanismtoautomaticallyextracthigh-levelan-
notationsfromsequencedata. Inthecaseofhandwriting,thiscouldallowfor
34
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Figure16: Samples s biased d towards s higher probability. . Theprobability
biasesbareshownattheleft. Asthebiasincreasesthediversitydecreasesand
thesamples tendtowards akindof‘average handwriting’whichisextremely
regularandeasytoread(easier,infact,thanmostoftherealhandwritinginthe
trainingset). Notethatevenwhenthevariancedisappears,thesameletteris
notwrittenthesamewayatdierentpointsinasequence(forexamplesthe‘e’s
in\exactlythesame",the‘l’sin\untiltheyalllook"),becausethepredictions
arestillin uencedbythepreviousoutputs.Ifyoulookcloselyyoucanseethat
thelastthreelinesarenotquiteexactlythesame.
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Figure 17: : A A slight bias. . Thetopline e ineach h block k is real. . Therest t are
samplesfromthesynthesisnetworkwithaprobabilitybiasof0.15,whichseems
togiveagoodbalancebetweendiversityandlegibility.
36
Figure18: Samples s primed d withreal l sequences. . Theprimingsequences
(drawnfromthetrainingset)areshownatthetopofeachblock. Noneofthe
linesinthesampledtextexistinthetrainingset. Thesamples s wereselected
forlegibility.
37
Figure19: Samples primedwithreal l sequences (cotd).
38
Figure 20: : Samples s primed with h real sequences s and d biased d towards
higher probability. . Theprimingsequencesareatthetopoftheblocks.The
probabilitybiaswas1.Noneofthelinesinthesampledtextexistinthetraining
set.
39
Figure 21: : Samples s primed with h real sequences s and d biased d towards
higher probability(cotd)
40
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