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ITMO RecSys course. Autumn 2014. Lecture 2

Andrey Danilchenko
Andrey Danilchenko

ITMO RecSys course. Autumn 2014. Lecture 2: SVD continued. Advanced SVD methods: iALS, ALS1, iALS1, rankALS. Explanations for SVD recommendations

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亠从仂仄亠仆亟舒亠仍仆亠 亳亠仄 亠从亳 2: SVD continued 仆亟亠亶 舒仆亳仍亠仆从仂 25 仂从磡 2014
弌从舒 Advanced SVD methods iALS ALS1, iALS1 rankALS Explanations for SVD
Advanced SVD methods iALS ALS1-optimization rankALS
Advanced SVD methods iALS
SVD revisited 仂亟亠仍: Tqi ruui =亮 + bu + bi + pu argmin p*q*b* 裡 2 ( r亮 b b pTq) uui u i u i (u,i )R 2 ( ) +了 pu 2 + qi 2 + bu 2 + bi 个仆从亳 仂亳弍从亳:
SVD revisited 仂亟亠仍: Tqi ruui =亮 + bu + bi + pu argmin p*q*b* 裡 2 ( r亮 b b pTq) uui u i u i (u,i )R 2 ( ) +了 pu 2 + qi 2 + bu 2 + bi 个仆从亳 仂亳弍从亳: 仂 仂 舒从仂亠 r ? uui
舒从 亳仗仂仍亰仂于舒 implicit feedback?
Implicit feedback prediction 亟亠: rating => (preference, confidence) pui {0,1} cui + (u, i) R pui =1 pui = 0 亳仆舒亠 cui =1+留rui 亳仍亳 cui =1+留 log 1+ rui ( 硫 )
Implicit SVD model argmin x*y* 裡 2 T yi ( ) cui pui xu (u,i ) 裡 + yi +了 xu 2 u 2 裡 i # $ % & ' ( 个仆从亳 仂亳弍从亳: 亅仂 于仂亟亳 从 舒于仆亠仆亳礆 亟仍 ALS: xu = (了 I +Y TCuY )1 Y TCu p(u) yi = 了 I + XTCi( X)1 XTCi p(i)
Implicit SVD model argmin x*y* 裡 2 T yi ( ) cui pui xu (u,i ) 裡 + yi +了 xu 2 u 2 裡 i # $ % & ' ( 个仆从亳 仂亳弍从亳: 亅仂 于仂亟亳 从 舒于仆亠仆亳礆 亟仍 ALS: xu = (了 I +Y TCuY )1 Y TCu p(u) yi = 了 I + XTCi( X)1 XTCi p(i) 仂 亠 仗仂弍仍亠仄舒!
丕从仂亠仆亳亠 (iALS) 亟亠: Y TCuY = Y TY +Y T (Cu I )Y 仗亳亠仄 于 C 于亠亞仂 仆亠仆仍亠于 仍亠仄亠仆仂于, u I nu 舒从 亢亠 于 C 于亠亞仂 仆亠仆仍亠于 仍亠仄亠仆仂于, up(u) nu 舒 Y 仆亠 亰舒于亳亳 仂 仗仂仍亰仂于舒亠仍! TY 仂亞仂: 仂弍仆仂于仍磳仄 于亠从仂舒 仗仂仍亰仂于舒亠仍亠亶 亰舒 O f 2N + f 3 ( U )
仆亳亳 iALS 仗亳亠仄 ALS-舒亞 于 仗仂亠仆仆仂亶 仂仄亠: xu = (了 I +Y TCuY )1 Y TCup(u) = 了 I + Au ( )1 du 于亠亟亠仄 束仆仍亠于仂亞仂損 仗仂仍亰仂于舒亠仍 弍亠亰 亳亟弍亠从舒: 裡 裡 T yd= cpyi 0 0 0i A0 = c0yi i i 丐仂亞亟舒 亟仍 仂舒仍仆 仗仂仍亰仂于舒亠仍亠亶 于于仂亟亳仄: T yi 裡 Au = A0 + (cui c0 )yi (u,i )N(u) 裡 du = d0 + cui pui c0 p0 ( ) yi (u,i)N(u)
仆亳亳 iALS 仗仂亟仂仍亢亠仆亳亠 舒从 于弍亳舒 c 亳 ? 0 p0
仆亳亳 iALS 仗仂亟仂仍亢亠仆亳亠 舒从 于弍亳舒 c 亳 ? 0 p0 舒从 亳 舒仆亠: (u, i) N pui =1 pui = 0 亳仆舒亠 cui =1+留rui
仆亳亳 iALS 仗仂亟仂仍亢亠仆亳亠 舒从 于弍亳舒 c 亳 ? 0 p0 舒从 亳 舒仆亠: (u, i) N pui =1 pui = 0 亳仆舒亠 cui =1+留rui p0 = 0 c0 =1
Advanced SVD methods ALS1-optimization
Ridge regression 仂亟亠仍: yi wT xi wTw0 argmin w yi wT xi ( )2 +了wTw n裡 i=1 # $ % & ' ( 个仆从亳 仂亳弍从亳: 丐仂仆仂亠 亠亠仆亳亠: w = (了 I + XTX)1 XT y = (了 I + A)1 d A = XTX d = XT y
ALS revisited 仂亟亠仍: Tqi ruui = pu argmin p*q*b* 裡 2 ( r pTq) uui u i (u,i )R 2 ( ) +了 pu 2 + qi 个仆从亳 仂亳弍从亳: P-step: pu = 了nuI + Au ( )1 du Au =Q[u]TQ[u] = qiqi T 裡 i:(u,i)R 裡 d =Q[u]T ru = ruiqi i:(u,i)R Q-step: qi = 了niI + Ai ( )1 di Ai = P[i]T P[i] = pupu T 裡 u:(u,i)R 裡 di = P[i]T ri = rui pu u:(u,i)R
Ridge regression (RR1 optimization) 仂亟亠仍: yi wT xi wTw0 argmin w n裡 了wTw+ wT xi yi ( )2 i=1 # $ % & ' ( 个仆从亳 仂亳弍从亳: 仂从仂仂亟亳仆舒仆亶 仗从 (仗仂 于亠仄 k): 裡 n wkxik yi wlxil i=1 lk wk 裡 n xe i=1 iki 了 + xxi=1 ikik n ( 裡 )
ALS1-仂仗亳仄亳亰舒亳 仄仂亟亠仍亳 SVD 仂亟亠仍: Tqi ruui = pu argmin p*q*b* 裡 2 ( r pTq) uui u i (u,i )R 2 ( ) +了 pu 2 + qi 个仆从亳 仂亳弍从亳: P-step: RR1 亟仍 pu Q-step: RR1 亟仍 qi
ALS1: 舒亰弍仂 P-舒亞舒 ei = yi +wT xi for k in 1..f: ei = ei +wkxik wk = 裡 n cxe i=1 iiki 了 + cxxi=1 iikik n ( 裡 ) ei = ei wkxik 仂亞仂: 仂弍仆仂于仍磳仄 仂亟仆仂亞仂 仗仂仍亰仂于舒亠仍 亰舒 仂仍仆仂亠 仂弍仆仂于仍亠仆亳亠 亰舒 O nu ( f ) O(Nf )
iALS1 亟亠: 于仄亠仂 束亳仆亠亳亠从仂亞仂損 仗仂仍亰仂于舒亠仍 弍亟亠仄 亳仗仂仍亰仂于舒 束亳仆亠亳亠从亳亶損 亳亟弍亠从 舒亢亟仂仄 仗仂仍亰仂于舒亠仍 仂仗仂舒于亳仄 亳亟弍亠从: positive: 仗仂舒于亳仍 ( p, c) uiui 于亠仄 q亟仍 i i N(u) negative (cancellation): 于亠仄 亟仍 negative (synthetic): 于亠仄 亟仍 regularization: j-亶 仂从亠 亟仍 p0,c0 ( ) qi i N(u) rk, c0 ( ) gk k 1..K (0,1) 了 I j 1..K
舒从 亳舒 束亳仆亠亳亠从亳亠損 于亠从仂舒? 舒亰仍仂亢亳仄 仄舒亳 仗仂 仂弍于亠仆仆仄 于亠从仂舒仄: 裡 T y= SST i A0 = c0yi i 仗亳亠仄 于 仄舒亳舒 亟亳舒亞仂仆舒仍仆舒 K K A0 = SST =GTG G = ST GT = S 于亠从仂舒 g 仂从亳 G 仆舒仍仂亞亳仆仂 d0 =GTr , 仂从亟舒 仗仂仍亳仄 亰仆舒亠仆亳 rk
Advanced SVD methods Rank ALS
仗亳仄亳亰亳仂于舒 舒仆亢亳仂于舒仆亳亠 亟仍 亰舒亟舒亳 舒仆亢亳仂于舒仆亳 仂仂仂!
Objective functions 裡 裡 fP () = cui rui rui ( )2 iI uU 裡 裡 裡 fR () = cui sj rui ruj ( ) rui ruj ( ) #$ 2 %& jI iI uU For prediction: For ranking (pairwise):
Training prediction fP () pu T pu rui ( )qi 裡 = = cui qi iI T 裡 = cuiqiqi iI & ' ( ) 裡 T = * +pu cuiruiqi iI = Au pu bu P-舒亞: 亠仄 舒弍仂: O f 2N + f 3 ( U )
Training ranking fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1
仂 仗仂仆磿?
Training ranking (for user) fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1
Training ranking (for user) fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1
Training ranking (for user) fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1 亠 亟于仂亶仆 亰舒于亳亳仄仂亠亶!
Training ranking (for item) fR (P,Q) qi 裡 裡 p+ u = cuisj pu jI T qi qj ( ) rui ruj ( ) %& '( uU 裡 裡 p= u + cujsi pu ( ) jI T qj qi ( ) ruj rui ( ) %& '( uU T 裡 = cui pu pu u ) * + , - . sj 裡 j ) * ++ , - .. T 裡 qi cui pu pu u ) * + , - . T sjqj 裡 j ) * ++ , - .. 裡 cui purui u ) * + , - . sj 裡 j ) * ++ , - .. 裡 裡 pu + cui sjruj j ) * ++ , - .. u ) * ++ , - .. 裡 T cujqj 裡 pupu j ) * ++ , - .. u ) * , .. ++ - 裡 T cuj 裡 si pupu j ) * ++ , - .. u ) * ++ , - .. siqi + 裡 裡 pu + cujruj j ) * ++ , - .. u ) * ++ , - .. 裡 裡 si puruj cuj j ) * ++ , - .. u ) * ++ , - .. si = + p1 p2 + p3si bsi ( ) = A!1 + Asi ( )qi A! q b!1
亠仗亠 亟仂弍舒于亳仄 亠亞仍亳亰舒亳
ITMO RecSys course. Autumn 2014. Lecture 2
Explanations
舒仄 仗仂仆舒于亳 Y! 仍亳 于舒仄 仆舒于亳 X, 仂 仗仂仗仂弍亶亠 Y
Item-based model explanation 仂亟亠仍: 裡 rui = wijrui k jNu 从仍舒亟 于 亠亶亳仆亞: wijrui 弍亠亠仄 j 仄舒从亳仄舒仍仆仄亳 于从仍舒亟舒仄亳 仂 亳 亠 仂弍仆亠仆亳亠! j
ALS explanation 仂亟亠仍: rui = yi T (了 I +Y TY )1 T xu = yi Y Tr (u) 裡 u ruj rui = sij (u,i)R 1 Wu = ( Y TY +了 I )su = yTWuyij i j 丐仂亞亟舒 仗亠亠仗亳亠仄 仄仂亟亠仍 从舒从 urj uj 从仍舒亟 于 亠亶亳仆亞: sij 弍亠亠仄 j 仄舒从亳仄舒仍仆仄亳 于从仍舒亟舒仄亳 仂 亳 亠 仂弍仆亠仆亳亠!
ALS explanation 仂亟亠仍: rui = yi T (了 I +Y TY )1 T xu = yi Y Tr (u) 裡 rui = Vuqi ( )T Vuqj ( )ruj (u,i)R 1 Wu = ( Y TY +了 I )su = yTWuyij i j 亠亠仗亳亠仄 仄仂亟亠仍 从舒从 Wu =Vu TVu Vu 丐仂亞亟舒 仄仂亢仆仂 舒仄舒亳于舒 从舒从 舒弍仍仂仆 仗仂仍亰仂于舒亠仍从仂亞仂 仄仆亠仆亳 仗仂 item-
iALS explanation 仂亟亠仍: pui = yi T (了 I +Y TCuY )1 T xu = yi Y TCup(u) 裡 u cpuj uj pui = sij pui>0 Wu = ( Y TCuY +了 I )1 su = yTWuyij i j 丐仂亞亟舒 仗亠亠仗亳亠仄 仄仂亟亠仍 从舒从 j ucp= sucuj uj ij uj 从仍舒亟 于 亠亶亳仆亞: sij 弍亠亠仄 j 仄舒从亳仄舒仍仆仄亳 于从仍舒亟舒仄亳 仂 亳 亠 仂弍仆亠仆亳亠!
仆亟亠亶 舒仆亳仍亠仆从仂 舒亰舒弍仂亳从 www.4ducks.ru

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ITMO RecSys course. Autumn 2014. Lecture 2

  • 1. 亠从仂仄亠仆亟舒亠仍仆亠 亳亠仄 亠从亳 2: SVD continued 仆亟亠亶 舒仆亳仍亠仆从仂 25 仂从磡 2014
  • 2. 弌从舒 Advanced SVD methods iALS ALS1, iALS1 rankALS Explanations for SVD
  • 3. Advanced SVD methods iALS ALS1-optimization rankALS
  • 5. SVD revisited 仂亟亠仍: Tqi ruui =亮 + bu + bi + pu argmin p*q*b* 裡 2 ( r亮 b b pTq) uui u i u i (u,i )R 2 ( ) +了 pu 2 + qi 2 + bu 2 + bi 个仆从亳 仂亳弍从亳:
  • 6. SVD revisited 仂亟亠仍: Tqi ruui =亮 + bu + bi + pu argmin p*q*b* 裡 2 ( r亮 b b pTq) uui u i u i (u,i )R 2 ( ) +了 pu 2 + qi 2 + bu 2 + bi 个仆从亳 仂亳弍从亳: 仂 仂 舒从仂亠 r ? uui
  • 8. Implicit feedback prediction 亟亠: rating => (preference, confidence) pui {0,1} cui + (u, i) R pui =1 pui = 0 亳仆舒亠 cui =1+留rui 亳仍亳 cui =1+留 log 1+ rui ( 硫 )
  • 9. Implicit SVD model argmin x*y* 裡 2 T yi ( ) cui pui xu (u,i ) 裡 + yi +了 xu 2 u 2 裡 i # $ % & ' ( 个仆从亳 仂亳弍从亳: 亅仂 于仂亟亳 从 舒于仆亠仆亳礆 亟仍 ALS: xu = (了 I +Y TCuY )1 Y TCu p(u) yi = 了 I + XTCi( X)1 XTCi p(i)
  • 10. Implicit SVD model argmin x*y* 裡 2 T yi ( ) cui pui xu (u,i ) 裡 + yi +了 xu 2 u 2 裡 i # $ % & ' ( 个仆从亳 仂亳弍从亳: 亅仂 于仂亟亳 从 舒于仆亠仆亳礆 亟仍 ALS: xu = (了 I +Y TCuY )1 Y TCu p(u) yi = 了 I + XTCi( X)1 XTCi p(i) 仂 亠 仗仂弍仍亠仄舒!
  • 11. 丕从仂亠仆亳亠 (iALS) 亟亠: Y TCuY = Y TY +Y T (Cu I )Y 仗亳亠仄 于 C 于亠亞仂 仆亠仆仍亠于 仍亠仄亠仆仂于, u I nu 舒从 亢亠 于 C 于亠亞仂 仆亠仆仍亠于 仍亠仄亠仆仂于, up(u) nu 舒 Y 仆亠 亰舒于亳亳 仂 仗仂仍亰仂于舒亠仍! TY 仂亞仂: 仂弍仆仂于仍磳仄 于亠从仂舒 仗仂仍亰仂于舒亠仍亠亶 亰舒 O f 2N + f 3 ( U )
  • 12. 仆亳亳 iALS 仗亳亠仄 ALS-舒亞 于 仗仂亠仆仆仂亶 仂仄亠: xu = (了 I +Y TCuY )1 Y TCup(u) = 了 I + Au ( )1 du 于亠亟亠仄 束仆仍亠于仂亞仂損 仗仂仍亰仂于舒亠仍 弍亠亰 亳亟弍亠从舒: 裡 裡 T yd= cpyi 0 0 0i A0 = c0yi i i 丐仂亞亟舒 亟仍 仂舒仍仆 仗仂仍亰仂于舒亠仍亠亶 于于仂亟亳仄: T yi 裡 Au = A0 + (cui c0 )yi (u,i )N(u) 裡 du = d0 + cui pui c0 p0 ( ) yi (u,i)N(u)
  • 13. 仆亳亳 iALS 仗仂亟仂仍亢亠仆亳亠 舒从 于弍亳舒 c 亳 ? 0 p0
  • 14. 仆亳亳 iALS 仗仂亟仂仍亢亠仆亳亠 舒从 于弍亳舒 c 亳 ? 0 p0 舒从 亳 舒仆亠: (u, i) N pui =1 pui = 0 亳仆舒亠 cui =1+留rui
  • 15. 仆亳亳 iALS 仗仂亟仂仍亢亠仆亳亠 舒从 于弍亳舒 c 亳 ? 0 p0 舒从 亳 舒仆亠: (u, i) N pui =1 pui = 0 亳仆舒亠 cui =1+留rui p0 = 0 c0 =1
  • 16. Advanced SVD methods ALS1-optimization
  • 17. Ridge regression 仂亟亠仍: yi wT xi wTw0 argmin w yi wT xi ( )2 +了wTw n裡 i=1 # $ % & ' ( 个仆从亳 仂亳弍从亳: 丐仂仆仂亠 亠亠仆亳亠: w = (了 I + XTX)1 XT y = (了 I + A)1 d A = XTX d = XT y
  • 18. ALS revisited 仂亟亠仍: Tqi ruui = pu argmin p*q*b* 裡 2 ( r pTq) uui u i (u,i )R 2 ( ) +了 pu 2 + qi 个仆从亳 仂亳弍从亳: P-step: pu = 了nuI + Au ( )1 du Au =Q[u]TQ[u] = qiqi T 裡 i:(u,i)R 裡 d =Q[u]T ru = ruiqi i:(u,i)R Q-step: qi = 了niI + Ai ( )1 di Ai = P[i]T P[i] = pupu T 裡 u:(u,i)R 裡 di = P[i]T ri = rui pu u:(u,i)R
  • 19. Ridge regression (RR1 optimization) 仂亟亠仍: yi wT xi wTw0 argmin w n裡 了wTw+ wT xi yi ( )2 i=1 # $ % & ' ( 个仆从亳 仂亳弍从亳: 仂从仂仂亟亳仆舒仆亶 仗从 (仗仂 于亠仄 k): 裡 n wkxik yi wlxil i=1 lk wk 裡 n xe i=1 iki 了 + xxi=1 ikik n ( 裡 )
  • 20. ALS1-仂仗亳仄亳亰舒亳 仄仂亟亠仍亳 SVD 仂亟亠仍: Tqi ruui = pu argmin p*q*b* 裡 2 ( r pTq) uui u i (u,i )R 2 ( ) +了 pu 2 + qi 个仆从亳 仂亳弍从亳: P-step: RR1 亟仍 pu Q-step: RR1 亟仍 qi
  • 21. ALS1: 舒亰弍仂 P-舒亞舒 ei = yi +wT xi for k in 1..f: ei = ei +wkxik wk = 裡 n cxe i=1 iiki 了 + cxxi=1 iikik n ( 裡 ) ei = ei wkxik 仂亞仂: 仂弍仆仂于仍磳仄 仂亟仆仂亞仂 仗仂仍亰仂于舒亠仍 亰舒 仂仍仆仂亠 仂弍仆仂于仍亠仆亳亠 亰舒 O nu ( f ) O(Nf )
  • 22. iALS1 亟亠: 于仄亠仂 束亳仆亠亳亠从仂亞仂損 仗仂仍亰仂于舒亠仍 弍亟亠仄 亳仗仂仍亰仂于舒 束亳仆亠亳亠从亳亶損 亳亟弍亠从 舒亢亟仂仄 仗仂仍亰仂于舒亠仍 仂仗仂舒于亳仄 亳亟弍亠从: positive: 仗仂舒于亳仍 ( p, c) uiui 于亠仄 q亟仍 i i N(u) negative (cancellation): 于亠仄 亟仍 negative (synthetic): 于亠仄 亟仍 regularization: j-亶 仂从亠 亟仍 p0,c0 ( ) qi i N(u) rk, c0 ( ) gk k 1..K (0,1) 了 I j 1..K
  • 23. 舒从 亳舒 束亳仆亠亳亠从亳亠損 于亠从仂舒? 舒亰仍仂亢亳仄 仄舒亳 仗仂 仂弍于亠仆仆仄 于亠从仂舒仄: 裡 T y= SST i A0 = c0yi i 仗亳亠仄 于 仄舒亳舒 亟亳舒亞仂仆舒仍仆舒 K K A0 = SST =GTG G = ST GT = S 于亠从仂舒 g 仂从亳 G 仆舒仍仂亞亳仆仂 d0 =GTr , 仂从亟舒 仗仂仍亳仄 亰仆舒亠仆亳 rk
  • 25. 仗亳仄亳亰亳仂于舒 舒仆亢亳仂于舒仆亳亠 亟仍 亰舒亟舒亳 舒仆亢亳仂于舒仆亳 仂仂仂!
  • 26. Objective functions 裡 裡 fP () = cui rui rui ( )2 iI uU 裡 裡 裡 fR () = cui sj rui ruj ( ) rui ruj ( ) #$ 2 %& jI iI uU For prediction: For ranking (pairwise):
  • 27. Training prediction fP () pu T pu rui ( )qi 裡 = = cui qi iI T 裡 = cuiqiqi iI & ' ( ) 裡 T = * +pu cuiruiqi iI = Au pu bu P-舒亞: 亠仄 舒弍仂: O f 2N + f 3 ( U )
  • 28. Training ranking fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1
  • 30. Training ranking (for user) fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1
  • 31. Training ranking (for user) fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1
  • 32. Training ranking (for user) fR (P,Q) pu 裡 qi qj ( )T 裡 = = cui sj jI pu rui ruj ( ) %&' ()* qi qj ( ) iI 裡 = sj j + , -- . / 00 T cuiqiqi 裡 i + , - . / 0 裡 pu cuiqi i + , - . / 0 T sjqj 裡 j + , -- . / 00 pu 裡 sjqj j + , -- . / 00 T cuiqi 裡 i + , - . / 0 T 裡 pu + cuiqiqi i + , - . / 0 T sjqjqj 裡 j + , -- . / 00 pu 裡 cuirui i + , - . / 0 sjqj 裡 j + , -- . / 00 裡 cui i + , - . / 0 sjqjruj 裡 j + , -- . / 00 = = !1 q!r r ! q + 1!b ( ) A q ! qT ! qqT + 1!A ( ) pu b!1 亠 亟于仂亶仆 亰舒于亳亳仄仂亠亶!
  • 33. Training ranking (for item) fR (P,Q) qi 裡 裡 p+ u = cuisj pu jI T qi qj ( ) rui ruj ( ) %& '( uU 裡 裡 p= u + cujsi pu ( ) jI T qj qi ( ) ruj rui ( ) %& '( uU T 裡 = cui pu pu u ) * + , - . sj 裡 j ) * ++ , - .. T 裡 qi cui pu pu u ) * + , - . T sjqj 裡 j ) * ++ , - .. 裡 cui purui u ) * + , - . sj 裡 j ) * ++ , - .. 裡 裡 pu + cui sjruj j ) * ++ , - .. u ) * ++ , - .. 裡 T cujqj 裡 pupu j ) * ++ , - .. u ) * , .. ++ - 裡 T cuj 裡 si pupu j ) * ++ , - .. u ) * ++ , - .. siqi + 裡 裡 pu + cujruj j ) * ++ , - .. u ) * ++ , - .. 裡 裡 si puruj cuj j ) * ++ , - .. u ) * ++ , - .. si = + p1 p2 + p3si bsi ( ) = A!1 + Asi ( )qi A! q b!1
  • 34. 亠仗亠 亟仂弍舒于亳仄 亠亞仍亳亰舒亳
  • 37. 舒仄 仗仂仆舒于亳 Y! 仍亳 于舒仄 仆舒于亳 X, 仂 仗仂仗仂弍亶亠 Y
  • 38. Item-based model explanation 仂亟亠仍: 裡 rui = wijrui k jNu 从仍舒亟 于 亠亶亳仆亞: wijrui 弍亠亠仄 j 仄舒从亳仄舒仍仆仄亳 于从仍舒亟舒仄亳 仂 亳 亠 仂弍仆亠仆亳亠! j
  • 39. ALS explanation 仂亟亠仍: rui = yi T (了 I +Y TY )1 T xu = yi Y Tr (u) 裡 u ruj rui = sij (u,i)R 1 Wu = ( Y TY +了 I )su = yTWuyij i j 丐仂亞亟舒 仗亠亠仗亳亠仄 仄仂亟亠仍 从舒从 urj uj 从仍舒亟 于 亠亶亳仆亞: sij 弍亠亠仄 j 仄舒从亳仄舒仍仆仄亳 于从仍舒亟舒仄亳 仂 亳 亠 仂弍仆亠仆亳亠!
  • 40. ALS explanation 仂亟亠仍: rui = yi T (了 I +Y TY )1 T xu = yi Y Tr (u) 裡 rui = Vuqi ( )T Vuqj ( )ruj (u,i)R 1 Wu = ( Y TY +了 I )su = yTWuyij i j 亠亠仗亳亠仄 仄仂亟亠仍 从舒从 Wu =Vu TVu Vu 丐仂亞亟舒 仄仂亢仆仂 舒仄舒亳于舒 从舒从 舒弍仍仂仆 仗仂仍亰仂于舒亠仍从仂亞仂 仄仆亠仆亳 仗仂 item-
  • 41. iALS explanation 仂亟亠仍: pui = yi T (了 I +Y TCuY )1 T xu = yi Y TCup(u) 裡 u cpuj uj pui = sij pui>0 Wu = ( Y TCuY +了 I )1 su = yTWuyij i j 丐仂亞亟舒 仗亠亠仗亳亠仄 仄仂亟亠仍 从舒从 j ucp= sucuj uj ij uj 从仍舒亟 于 亠亶亳仆亞: sij 弍亠亠仄 j 仄舒从亳仄舒仍仆仄亳 于从仍舒亟舒仄亳 仂 亳 亠 仂弍仆亠仆亳亠!