Probabilistic Latent Semantic Analysis


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b

(b)

(a) P(z|d)

P(d)

d

z

P(z)

P(w|z)

w

d

P(d|z)

z

P(w|z)

w

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:Û „CÜ Š {‹ t©Û xw i* Ü ‰ y #Û * evÜ tÛ i*w * e;Ü

Û éyÜ Œ'{ tÛ iyŽ Ü ‰ y y #$Û } evÜ tÛ i*w } eÜ Û yÜ Š {‹ Œ'{l €d$|n*h9d%Cr=w´q$pJw[w[r=kE¶£|pEh[i ‡ d hFgˆ}=¶*n*h9r=i ‡ jYrÅquh9d µ kEd jYd kCi[wml>°od °ar=}=}Äw9i[pUCzi ‡ dçh[d }7g-i9r=n*kEw ‡ r=fŒAd i{°˜dd k i ‡ dçfEh[nyfUnyw[dM j%nECd}§gˆkUº>`C†»r=kj%nyh[deCd i$g-r=}{‹

embedding simplex spanned sub-simplex



+ P(w|d) KL divergence projection

P(w|z 2) P(w|z 3)

P(w|z 1) 0

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LSA

1000 900 800

1500 EM

700

PLSA

1000

600 TEM

200 400 600 800 Latent space dimensions

1000

500

LSA PLSA 500 1000 1500 Latent space dimensions

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ÉukEd;n-|Fi ‡ dĂyd zfEh[nyŒE}=d j%wYr=kÃr=kC|ªn*h[jg-i9rtnyk¹h9d i[h9rtdv*g-}r=w ›-ÔyŸ[™-¨›mŸ{œ¡œ1}¥ h*¤ ՜1¥CÓ ° ‡ r7q ‡Ä‡ gˆwrti9wejg-r=k„g-fEfE}=r7q gˆi[r=n*kr=k ­ypEd h9z*²{ŒUgˆw[dMh9d i[h9r=d v*g-}{‹^¾ ‡ d¼jYn*w9i˜fAn*fEpJ}Ågˆho| g-jYr=}tz"n-|^r=kC² |nyh[j‰gˆi[r=n*kh9d i[h9rtdv*g-}^i9dMq ‡ kEr7­ypEd w r=wŒUg-w9dM;n*k;i ‡ d U¤$¡$Ÿ[™-— ¦*æE›*¡$¤ Ä™ hy¤ CÛª½¼`EÈcÜ|n*h^CnEq$pJj%d kCi[waß=ãMäâ[‹ ad h9d*lC°od ‡ gmvyd pEi9rt}=r=Æ dMgh$g-i ‡ d hw[i[h$g-r=¶ ‡ iÖ|n*h9°4gˆh´h9d fEh9d w[d kCi$g-i[r=nykČUgˆw[dM n*ki ‡ dÛpJkOi9h´gˆkEw{|nyh[jYdMEÜi[dh[j |h[dM­ypEdkUq$r=d Dw #Û * eÜ"i[n-² ¶*di ‡ d hc°arti ‡ i ‡ dw[i$g-kUJg-h´»qsn*w[r=kEdjg-i$q ‡ rtkJ¶|pEkAq$i[r=n*kl g;j%nyh[d„Cd i$g-r=}=dMd$·EfAd h[r=jYd kCi´g-}egˆkUg-}=zCw[r=w%q gˆk¯ŒAd|n*pJkU r=k¯(ß â֋Y¾ ‡ dw$g-j%dh[dfEh[d w9d kCi´gˆi[r=n*k¯g-fJfE}=rtdw i[n„­ypEd h9rtdw Cl w[ni ‡ g-ii ‡ d„jg-i´q ‡ r=kE¶£|pEkUq$i9r=n*k®|n*hi ‡ dŒUg-w9d }=r=kEd„i[dh[j j‰gˆi´q ‡ r=kE¶ jYd i ‡ n³Yq g-kŒAd °ah[r=i9i[d k„g-w … #Û * evÜ #Û eÜ Û$*

-Ü'  … #Œ $Û } eÜ U  … #Û eÜ U Û$ã yÜ Œ Œ …{kúg-i9d kCiF`Cd jg-kCi[r7q…{kUCds·³r=kE¶¸Û º>`³…[Ü li ‡ d%n*h9r=¶*r=kUg-}v*dMq9² i[nyh^w[fAg*q$dh[dfEh[d w9d kCi´gˆi[r=n*kYn-|CnEq$pEjYd kCi[w^r=w^h9d fE}7g*qsdMŒCzg h[dfEh[d w9d kCi´gˆi[r=n*kYrtk"i ‡ da}=n-°o²[Cr=j%d kEw9r=n*kUgˆ}-}Ågˆi[d kCiw[fAg*q$degˆkU i ‡ d%w[r=jYrt}7gˆh[r=i{z%r=wFq$nyj%fEpJi[dMcŒUg-w9dMnyki ‡ gˆi h[dfEh[d w9d kCi´gÕ² i[r=nyk>‹ upEdh[r=d won*h^CnEq$pEjYd kCi[w^° ‡ r7q ‡ °˜d h9dekEnyi^fUgˆh[in-|+i ‡ d n*h9r=¶*r=kUg-}+q$ny}t}=dMqsi[r=n*kq g-kYŒUd M™ˆ h*¤ h!œ7¥ ŒCz‰g"w[r=jYfE}=daj‰g-i9h[r~· jpE}=i[r=fE}=r7q gˆi[r=n-k%Û{q9|´‹AßèámâA|n*haCd i$g-r=}=w´Ü ‹…{kYn*pEhod$·EfAd h[r=j%dkOi9wMl °od ‡ gmv*dag*q$i9pUg-}=}=zFq$nykEw[r7Cdh[dM!}tr=kEdmg-h^q$n*jŒJrtkAg-i[r=nykEw+n-|+i ‡ d n*h9r=¶*r=kUg-}w[r=j%r=}7g-h9rti{zw$q$n*h9dÄÛ$ã yÜÛª°˜d r=¶ ‡ i
Üg-kU¸i ‡ dnykEd Cd h9r=v*dM¸|h9n*ji ‡ d}7g-i9d kCiFw9fUg*qsd‰h9d fEh9d w[d kCi$g-i[r=nykÂÛª°˜drt¶ ‡ i ã
Ü ‹ ¾ ‡ duw´g-jYdar7CdMg-w ‡ gmvydaŒUd dk£gˆfEfE}=r=dMr=k‰ïh9n*ŒUgˆŒEr=}=rtw9i[r7qoº>gm² i[dkOi;`Cd j‰gˆkCi[r7q„…{kACd$·Er=kE¶×Û ï^º`C…[Ür=kq$n* k o$pEkUq$i9r=n*k×°arti ‡ i ‡ då>`C†×j%nECd }{‹>È£n*h[dfEh9dMq$r=w[d}tzClCi ‡ de}=n-°o²Ö³rtjYd kEw9rtnykUg-} h[dfEh[d w9d kCi´gˆi[r=n*kr=ki ‡ d M›*¡sŸ[™-—ž{æE›*¡$2¤ tÛ i*w OÜgˆkU t$Û i}w -Ü 9



A







$#



$#





90

70

60

50

45

80 60

MED

CRAN

CACM

50

CISI 40

70 50

35

60

40

precision [%]

30 40

50

30 40

25

30 20

30

20 15

20 20

10 10 10 10

cos LSI PLSI*

cos LSI PLSI*

0

cos LSI PLSI*

0 0

50 recall [%]

100

0 0

50 recall [%]

100

cos LSI PLSI*

5

0 0

50 recall [%]

100

0

50 recall [%]

100

‰j gˆi´q ‡ r=kE¶UlCº>`³… lEg-kUï^º`C… n*ki ‡ dE„i[d w9i¼q$ny}t}=dMqsi[r=n*kEwm‹ ‡ gmvyd¼ŒAd d kcpEi[r=}=r=Æ dMYi[nd v*gˆ}=pUg-i9d¼w9r=j%r=}7g-h[r=i9rtdwM‹+¾n%g*q ‡ r=d vyd i ‡ rtwmlE­ypEd h[r=d w ‡ gmv*dei9n ŒAde|ªn*}7CdM!rtkl*° ‡ r7q ‡ r=waCn*kJder=ki ‡ d ï^º`C†ŒOz µ ·Er=kE¶ui ‡ dptÛ2w i*ÜAfUgˆh´g-jYd i[dh[w^g-kU"q g-}7q$pE}7gˆi[r=kE¶ °od r=¶ ‡ i[wZtÛ$i}w -܁ŒOzY¾e]^È‹ ÉukEdÃgyCv*g-kCi$g-¶*d;n-|pEw9rtkJ¶¯w[i$g-i[r=w9i[r7q g-}j%nECd}twYvCwM‹`C½¼b i[dmq ‡ kEr7­*pJd w r=wei ‡ g-ir=i¼gˆ}t}=n-°awapEwi9n%w[zCw9i[d jg-i[r7q gˆ}=}tzq$n*j!² ŒEr=kEdCr~Þ§dh[d kCi;j%nECd }=wM‹B` ‡ r=}=dÄi ‡ r=ww ‡ nypE}7¯nyfEi[r=j‰gˆ}=}tz ŒAd‰CnykEd‰gyq q$n*h$Cr=kE¶;i[ng„€˜gmzyd w[r7gˆk¹j%nECd}˜q$n*jŒJrtkAg-i[r=nyk w´q ‡ dj%d*l+°od ‡ gmvyd pEi9rt}=r=Æ dM;gjpAq ‡ w9rtjYfE}=d heg-fJfEh[nCg*q ‡ r=k n*pJh d$·EfUdh[r=j%d kCi9w ° ‡ r7q ‡Ä‡ gˆw kEd vyd h[i ‡ d }=d w9ww ‡ n-°ak;d$·Uq$d }~² }=d kCi%fAd h{|ªn*h[jg-kUqsdÄg-kU h[n*ŒJpEw[i9kEd w[wm‹ -ug-j%d}tzCl^°od ‡ gmvyd w[r=jYfE}=zåqsn*jŒEr=kEdM"i ‡ deq$n*w9rtkJdaw´q$nyh[d wanˆ|ìgˆ}t}Jj%nECd }=w°ar=i ‡ g pEkEr~|nyh[j»°od r=¶ ‡ iM‹¾ ‡ dah9d w[pE}=i9rtkJ¶¼jYd i ‡ n³årtwh[d$|dh[h[dmi[n gˆw ï^º`C… ‹^]Ýj%fEr=h[r7q gˆ}=}tz°od ‡ gmv*dà|n*pJkUi ‡ dufUdh{|nyh[j‰gˆkUq$dai9n ŒAd%v*dh[z;h[n*ŒJpEw[i° ‹ hM‹ iM‹Cr~Þ§dh[d kCiYÛkEnykC² pJkEr~|n*h9j‰Üa°od r=¶ ‡ i9w g-kA‰gˆ}=w[n° ‹ hm‹ iM‹yi ‡ d
²{°˜d r=¶ ‡ iopEw9dMr=k‰q$nyjŒEr=kUg-i9r=n*k°arti ‡ i ‡ dFnyh[r=¶*r=kUgˆ}q$nyw[r=kEdw´q$nyh[d*‹¾ ‡ r=wer=w CpEdi[nYi ‡ dkJn*r=w[d h9d$² i[rr=r=ny¶*k%pEh9n-d |i J‡ Úed"ïrtd kCh[vyfEd }=h[d$w9·Ed¼r=i{i9z„d j%g-kUfA;d h$g-gmv*i[pJdh´h[dg-¶y;d |fEn*h9hadMq$g-r=kw[r=n*g-kw9fUg-dMw qsieg©j%|pEnEkUCq9d² } CpUqsrtkJ¶¼ŒAd kEd µ i[won-|>Ûªj%nECd }7Ügmv*d h$g-¶*r=kE¶A‹<-anyi[r7q$dai ‡ g-i^º>`³† h[dfEh[d w9d kCi´gˆi[r=n*kEwÝ|n*hCr~Þ§d h9d kCi OÇ|nyh[jgàkEd w[i9dMw[dm­*pJd kUq$d*l °ar=i ‡ OK[„CêÛª}tds|i´ÜagˆkUG O ãmä*ê!Ûh[r=¶ ‡ i$Ü ‹ ° ‡ r7q ‡ r=wkEn*iui[h[pJd |nyhi ‡ dw[i$g-i[r=w9i[r7q g-}jYn³³d }=wa° ‡ r7q ‡ g-h9d d$·EfAdMq$i[dm;i[n%q gˆfEi[pEh9d©g}7gˆh[¶*dhev*g-h9r=d i{z%nˆ|Ýh9dMg-w9n*kUgˆŒE}=dFCd$² i[r=nykUÜ ‹a¾ ‡ dFq$n*kACd kEw9dM„h9d w[pJ}ti9wrtkci[d h9j%wan-|agmv*d h$g-¶*dfEh9d$² q$nyj%fAn*w[r=i9rtnykEwM‹ q$r=w[r=nykFh9dMq g-}=}+Û{g-ii ‡ d ah[dmq g-}=}E}=d vyd }=weã  Ü+gˆh[dow[pEj!² ‰ j gˆh[r=Æ dM!r=k‰¾^gˆŒE}=d©ãylE° ‡ r=}=dei ‡ dFq$nyh[h9d w[fAn*kU³rtkJ¶‰fJh[dMq$r=w9rtnyk ‡ ‡ `d gmvyd^pEi9rt}=r=Æ dMåi d|n*}=}=n-°ar=kE¶|n*pEhj%dM³rtpJj²{w[r=Æ dMuw[i$g-kC² [ h m d q ‡ Egˆh´¹³n³qspEj%d kCiq$n*}=}=dMq$i9r=n*k¹°ar=i h[d}tdv*g-kUq$d„gˆw[w[dw[w[jYd kCiMÚ g-}=}qspEh[vyd w¼q gˆkŒUd|ªn*pEkU!r=dk rr=¶*pJh[d E‹ ad h9d©gˆh[dw[nyj%d Ûr7ÜoȄ]^b Û´ã yá*á ³n³qspEj%d kCiegˆŒEw[i9h´g*qsi[wo|h9n*jÑi ‡ d -eg-i9r=n*kUgˆ} g*JCr=i[r=n*kUgˆ}eCd i$g-r=}=w‰nˆ| i ‡ d;d$·EfUdh[r=j%d kCi$g-}w[d i9pEf>ڐï^º>`³† º+r=ŒEh$g-h9z%n-|aÈdMCr7q$r=kEdMÜlÛr=r7Üa_ a† -ÇÛ$f㠄 CnEq$pEjYd kCieg-ŒC² j%nECd}twog-pi O ×á* ä J„Cê „} $ê ãMäyê ‡ gmv*duŒUdd ki[h´gˆr=kEdMŒCz w[i9h´gyq$i[w nykÄg-dh[n*kAg-pEi9rÅqswe|ªh[n*jÒi ‡ d_oh´g-k µ d }7…{kEw9i[r=i[pEi9dn-| ¾e]^È |nyhdMg*q ‡ Egˆi´gYw[d i°ar=i ‡ ã  ‡ d }7y²{n*pEiFJg-i´gJ6‹ rEnyh ¾dMq ‡ kEny}tny¶*zEÜ lYÛªrtr=r7܄_o†e_aÈ Û á*ä „»g-ŒEw9i[h$g*q$i9w|h9n*ji ‡ d ï^º`C…+°˜dh[d fAn*h9ii ‡ d^ŒAd w[ih9d w[pE}=in*ŒJi´g-r=kEdmeŒCz g-kCzun-|Ui ‡ d w[d _o†e_a
È *n*pJh[kUgˆ}ÅÜlgˆkU¸Ûr=vEܼ_o…[`³…FÛ$f㠄  %gˆŒEw[i9h´gyq$i[wr=kÄ}=r~² j%nECd}twml-|ªn*heº>`³…^°˜dh9d fUnyh[iui ‡ dŒAd w[iuh[d w9pE}=ien*ŒJi´g-r=kEdm|nyh ŒEh$g-h[zw´qsrtdkUq$du|h[nyjçi ‡ da…{kEw9i[r=i[pJi[du|n*h^`EqsrtdkOi9r µ qa…{kC|ªn*h[jgm² i ‡ d‰nyfEi[r=jg-}^Cr=j%d kJw[r=n*k®Ûd$·EfE}=nyh[r=kE¶á*äm±EéEãm䉍³rtjYd kEw9rtnykEw g-iguw[i[dfFw9rtÆdan-|Uê*Ü‹>¾ ‡ daq$nyjŒEr=kUg-i9rtnyke°od r=¶ ‡ i
°ar=i ‡ i ‡ d rr=¶*pJh[dEÚ^ïh9dMq$r=w[r=nykC² h9dMq gˆ}t}q$pEh9v*d wu|nyh˜i9d h[j

2000

2000

K=48

1600 1400 1200

1600 1400 1200

0.6

0.7

0.8 beta

0.9

1

0.6

70

0.7

K=48

60 50 40 30

50 40

0.7

0.8 beta

0.9

1

0.6

0.7

$# 2 #

+

$#

#

$#

#

#

1

K=128

60



# #

0.9

30

0.6

"

0.8 beta

70

average precision

average precision

K=128

1800

perplexity

perplexity

1800

0.8 beta

0.9

1

q$nyw[r=kEd%ŒAg-w[d}tr=kEdYw´q$nyh[d ‡ gˆw ŒUdd k¯q$nOgˆh[w[d}tznyfEi[r=jYrtÆdM„ŒCz ‡ gˆkU+lEȄ]^blA_ a† -"Ú
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