Introduction to Low-rank Matrix Completion

행렬완성개론
IntroductiontoMatrixCompletion
KAIST 전산학부기계학습및지능연구실(MLILAB)
석사과정이준영
2018-07-18

“설명하기에 앞서, 이 슬라이드는 간결한 설명을 위해
엄밀하지 않게 제작되었음을 알려드립니다.”

파레토법칙과롱테일법칙
출처:http://mbanote2.tistory.com/197
추천의중요성
• IT서비스는오프라인점포와달리롱테일에있는상품들을진열하는데드는비용이매우적음.
• 심지어매입가격도훨씬저렴함. 
(음악스트리밍서비스의경우,신곡보다오래된곡이비용이더저렴함)
• 고객들에게고정된금액을받는데,고객만족도를유지하면서저렴한제품을공급할수있다면?

롱테일법칙사례
추천을통한판매가전체판매의35%https://m.blog.naver.com/PostView.nhn?blogId=sanny0314&logNo=220630765408&proxyReferer=https%3A%2F%2Fwww.google.co.kr%2F
추천의중요성
추천을통한재생이전체재생의67%http://www.hani.co.kr/arti/PRINT/694128.html
DiscoverWeekly추천을통한재생이전체재생의4%https://jacesky1.wordpress.com/2016/10/09/%EB%8D%B0%EC%9D%B4%ED%84%B0%EB%A1%9C-%EC%9D%8C%EC%95%85%EC%9D%84-%EB%8D%94%EC%9A%B1-
%EB%B9%9B%EB%82%98%EA%B2%8C-spotify/https://brunch.co.kr/@hmin0606/7
간단한알고리즘의도입만으로클릭률150%달성!KAIST리크루팅에서이야기했음.

롱테일법칙사례
추천을통한판매가전체판매의35%https://m.blog.naver.com/PostView.nhn?blogId=sanny0314&logNo=220630765408&proxyReferer=https%3A%2F%2Fwww.google.co.kr%2F
추천의중요성
추천을통한재생이전체재생의67%http://www.hani.co.kr/arti/PRINT/694128.html
DiscoverWeekly추천을통한재생이전체재생의4%https://jacesky1.wordpress.com/2016/10/09/%EB%8D%B0%EC%9D%B4%ED%84%B0%EB%A1%9C-%EC%9D%8C%EC%95%85%EC%9D%84-%EB%8D%94%EC%9A%B1-
%EB%B9%9B%EB%82%98%EA%B2%8C-spotify/https://brunch.co.kr/@hmin0606/7
간단한알고리즘의도입만으로클릭률150%달성!KAIST리크루팅에서이야기했음.
오늘세미나에서는
NETFLIX의추천에대해서
집중적으로다뤄보고자합니다!

넷플릭스:세계최대VOD스트리밍사이트
NetflixPrize
는한국형넷플릭스!

2006년,넷플릭스는NetflixPrize를개최
NetflixPrize
백만달러(약12억원)의상금이걸린대회를열었음!
현재넷플릭스추천엔진성능의10%를개선한추천엔진을만들면백만달러를주는대회.

2006년이후,NetflixPrize관련논문수폭발적증가
NetflixPrize
#Paperssearchedbykeyword"MatrixCompletion"inGoogleScholar
NetflixPrize를기점으로
많은연구자들이NetflixProblem을풀기위해노력하기시작!

당시넷플릭스가보유하고있던추천엔진
NetflixPrize
• 지금은Netflix에별점을주는기능이사라졌지만,이때만해도별점을주는기능이있었음.
• 그리고,그별점이력을바탕으로추천을수행했음.
• 기존넷플릭스의추천엔진:Cinematch
• "straightforwardstatisticallinearmodelswithalotofdataconditioning”
• Cinematch를10%개선하는것이얼마나어려울까?
• 10%개선에3년이걸림!
출처:https://web.archive.org/web/20070821194257/http://www.netflixprize.com/faq
출처:인터넷속의수학

TheProblemofNetflixPrize
다음과같은Rating행렬이주어진다고가정(열이유저,행이영화).
NetflixPrize
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
유저
영화

Rating행렬에서일부Rating에대한정보를알고있을때,
NetflixPrize
1 2 3 4 5 6 7 8 9 10
1 4 2 4
2 3 3 1 3 3
3 3 2 3 1 4
4 2 4 1 2 5
5 3 3 1
6 2 3 2
유저
영화

모르는행렬entry에대한rating을맞추는문제!
NetflixPrize
1 2 3 4 5 6 7 8 9 10
1 4 2 4
2 3 3 1 3 3
3 3 2 3 1 4
4 2 4 1 2 5
5 3 3 1
6 2 3 2
유저
영화
행렬을완성한다고해서행렬완성(MatrixCompletion)문제라고합니다.
*실제NetflixProblem에서는 rating이된timestamp정보까지주어졌긴했습니다만,간결함을위해이슬라이드에서는제외하였습니다.

NetflixPrizeWinner
NetflixPrize
In2009,theBellKor’sProgramaticChaos팀이NetflixPrize에서우승!
어떻게기존알고리즘을10%개선했을까?

Winner’sMainIdea
NetflixPrize
10%향상중에8%향상에기여한아이디어에대해서만소개!(나머지향상은대부분앙상블!)
1 2 3 4 5 6 7 8 9 10
1 4 2 4
2 3 3 1 3 3
3 3 2 3 1 4
4 2 4 1 2 5
5 3 3 1
6 2 3 2
유저
영화
아무런가정이없다면,행렬을완성할수없음!
(가정이없으면Gain도없다)
->Low-rank가정을추가해행렬을완성!

Recall:행렬의rank란무엇인가?
-정의:행렬A의rowspace의dimension과columnspace의dimension중작은값.
Low-rankMatrixCompletion(LRMC)

Recall:행렬의rank란무엇인가?
어떤행렬M의rank가k이면,
위등식을만족하는행렬P,Q가존재합니다.*
M P Q= x
mxn mxk kxn
*HowardAnton,RobertC.Busby.Theorem8.6.1.ContemporaryLinearAlgebra.ISBN978-0-471-16362-6Wiley.
-정의:행렬A의rowspace의dimension과columnspace의dimension중작은값.

-rank가행렬의크기에비해충분히작다는것!
low-rank란무엇인가?
k<<min(n,m)
M P Q= x
mxn mxk kxn

low-rank란무엇인가?
M P Q= x
mxn mxk kxn
넷플릭스Prize우승팀의경우에는k를약20정도로설정함!
m>48000,n>17000
k<<min(n,m)
-rank가행렬의크기에비해충분히작다는것!

Low-rankMatrixCompletionforNetflixPrize
M P Q≒ x
mxn mxk kxn
유저
m>48,000
영화
n>17,000
특성
k≒20
User1’s특성vector
Movie1’s특성vector

Low-rankMatrixCompletionforNetflixPrize
M P Q≒ x
mxn mxk kxn
유저
m>48,000
영화
n>17,000
특성
k≒20

왜Low-rank가정이합리적인가?(1/2)
M P Q≒ x
행렬M의rank가k라는말은,행렬M의각rowvector들이
행렬M의k개의rowvector들의linearcombination으로표현가능하다는말!

M P Q≒ x
다시말해서,어떤유저의취향은
소수(k명의)유저들의취향의linearcombination으로표현가능!
즉,서로서로취향이비슷하다!
행렬M의rank가k라는말은,행렬M의각rowvector들이
행렬M의k개의rowvector들의linearcombination으로표현가능하다는말!

M P Q≒ x
유저벡터p1
영화벡터q1

M P Q≒ x
유저벡터p1
영화벡터q1
액션 스릴러 드라마…
3
0.1
4 5
0.2 0.05
-고전추천시스템에서는데이터사이언티스트들이다음과같이추천을진행!
데이터사이언티스트들이유저마다,영화마다이값을정해서입력해줬어야했다.
유저벡터p1
영화벡터q1

M P Q≒ x
3
0.1
4 5
0.2 0.05
장르1 …장르2 장르k
-우리는각각의장르를latentvariable로볼수있다.
유저벡터p1
영화벡터q1
유저벡터p1
영화벡터q1

Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Setofknownentries
P⌦(S) =
⇢
Sij if (i, j) 2 ⌦,
0 otherwise.<latexit sha1_base64="DQjr8Uh3EhviIVyXrCXgxDxtHgk=">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</latexit><latexit sha1_base64="DQjr8Uh3EhviIVyXrCXgxDxtHgk=">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</latexit><latexit sha1_base64="DQjr8Uh3EhviIVyXrCXgxDxtHgk=">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</latexit><latexit sha1_base64="DQjr8Uh3EhviIVyXrCXgxDxtHgk=">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</latexit>

Setofknownentries
그러니까기존데이터를가장잘설명하는Low-rankMatrix를찾는것!
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
<latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">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</latexit><latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">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</latexit><latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">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</latexit><latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">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</latexit>
P⌦(S) =
⇢
Sij if (i, j) 2 ⌦,

Setofknownentries
아쉽게도이문제는NP-Hard!
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
P⌦(S) =
⇢
Sij if (i, j) 2 ⌦,

Setofknownentries
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
P⌦(S) =
⇢
Sij if (i, j) 2 ⌦,
아직은polynomialtime안에해를찾을수없다!

Setofknownentries
그러면,어떻게풀까?
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
P⌦(S) =
⇢
Sij if (i, j) 2 ⌦,

Setofknownentries
그러면,어떻게풀까?
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
P⌦(S) =
⇢
Sij if (i, j) 2 ⌦,
SVDMethod.

Recall:SingularVectorDecomposition(SVD)이란?
SVDMethod
-SVD:어떤행렬 을 로분해하는방법!M = U⌃V T
<latexit sha1_base64="ZFjjQ8DUS/zk3wWJONFQk3+fz0I=">AAAB+HicbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YsXoWLTFtpYNttNu3Q3CbubQgn9J148qHj1p3jz37htc9DWBwOP92aYmRcknCntON9WYWV1bX2juFna2t7Z3bP3DxoqTiWhHol5LFsBVpSziHqaaU5biaRYBJw2g+Ht1G+OqFQsjup6nFBf4H7EQkawNlLXtu/RNfJQ55H1BUaNp3rXLjsVZwa0TNyclCFHrWt/dXoxSQWNNOFYqbbrJNrPsNSMcDopdVJFE0yGuE/bhkZYUOVns8sn6MQoPRTG0lSk0Uz9PZFhodRYBKZTYD1Qi95U/M9rpzq89DMWJammEZkvClOOdIymMaAek5RoPjYEE8nMrYgMsMREm7BKJgR38eVl4p1Vriruw3m5epOnUYQjOIZTcOECqnAHNfCAwAie4RXerMx6sd6tj3lrwcpnDuEPrM8ffZ2R0w==</latexit><latexit sha1_base64="ZFjjQ8DUS/zk3wWJONFQk3+fz0I=">AAAB+HicbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YsXoWLTFtpYNttNu3Q3CbubQgn9J148qHj1p3jz37htc9DWBwOP92aYmRcknCntON9WYWV1bX2juFna2t7Z3bP3DxoqTiWhHol5LFsBVpSziHqaaU5biaRYBJw2g+Ht1G+OqFQsjup6nFBf4H7EQkawNlLXtu/RNfJQ55H1BUaNp3rXLjsVZwa0TNyclCFHrWt/dXoxSQWNNOFYqbbrJNrPsNSMcDopdVJFE0yGuE/bhkZYUOVns8sn6MQoPRTG0lSk0Uz9PZFhodRYBKZTYD1Qi95U/M9rpzq89DMWJammEZkvClOOdIymMaAek5RoPjYEE8nMrYgMsMREm7BKJgR38eVl4p1Vriruw3m5epOnUYQjOIZTcOECqnAHNfCAwAie4RXerMx6sd6tj3lrwcpnDuEPrM8ffZ2R0w==</latexit><latexit sha1_base64="ZFjjQ8DUS/zk3wWJONFQk3+fz0I=">AAAB+HicbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YsXoWLTFtpYNttNu3Q3CbubQgn9J148qHj1p3jz37htc9DWBwOP92aYmRcknCntON9WYWV1bX2juFna2t7Z3bP3DxoqTiWhHol5LFsBVpSziHqaaU5biaRYBJw2g+Ht1G+OqFQsjup6nFBf4H7EQkawNlLXtu/RNfJQ55H1BUaNp3rXLjsVZwa0TNyclCFHrWt/dXoxSQWNNOFYqbbrJNrPsNSMcDopdVJFE0yGuE/bhkZYUOVns8sn6MQoPRTG0lSk0Uz9PZFhodRYBKZTYD1Qi95U/M9rpzq89DMWJammEZkvClOOdIymMaAek5RoPjYEE8nMrYgMsMREm7BKJgR38eVl4p1Vriruw3m5epOnUYQjOIZTcOECqnAHNfCAwAie4RXerMx6sd6tj3lrwcpnDuEPrM8ffZ2R0w==</latexit><latexit sha1_base64="ZFjjQ8DUS/zk3wWJONFQk3+fz0I=">AAAB+HicbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YsXoWLTFtpYNttNu3Q3CbubQgn9J148qHj1p3jz37htc9DWBwOP92aYmRcknCntON9WYWV1bX2juFna2t7Z3bP3DxoqTiWhHol5LFsBVpSziHqaaU5biaRYBJw2g+Ht1G+OqFQsjup6nFBf4H7EQkawNlLXtu/RNfJQ55H1BUaNp3rXLjsVZwa0TNyclCFHrWt/dXoxSQWNNOFYqbbrJNrPsNSMcDopdVJFE0yGuE/bhkZYUOVns8sn6MQoPRTG0lSk0Uz9PZFhodRYBKZTYD1Qi95U/M9rpzq89DMWJammEZkvClOOdIymMaAek5RoPjYEE8nMrYgMsMREm7BKJgR38eVl4p1Vriruw3m5epOnUYQjOIZTcOECqnAHNfCAwAie4RXerMx6sd6tj3lrwcpnDuEPrM8ffZ2R0w==</latexit>
MAS109선대개시간에배움!M<latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit><latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit><latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit><latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit>
*여기서U,V의columnvectors는orthonormal basis여야하지만,논지에중요치않아넘어가도무방합니다.
M U
VT
=
mxn mxk
kxn
Σ
kxk
만약행렬M의rank가k라면,아래와같이분해된다!

Recall:SingularVectorDecomposition(SVD)이란?
SVDMethod
-SVD:어떤행렬 을 로분해하는방법!M = U⌃V T
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MAS109선대개시간에배움!M<latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit><latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit><latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit><latexit sha1_base64="2bCVaGjBVkw40Ey6VQ8hhrpoyIg=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL16EFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3Uz95hMqzWN5b8YJBhEdSN7njBorNe665YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldp2nUYQjOIZT8OACanALdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucPE56MpQ==</latexit>
*여기서U,V의columnvectors는orthonormal basis여야하지만,논지에중요치않아넘어가도무방합니다.
M U
VT
=
mxn mxk
kxn
Σ
kxk
만약행렬M의rank가k라면,아래와같이분해된다!
여기서핵심은행렬Σ는대각행렬이라는것!
Σij =0ifi!=j
그리고,Σii >=Σjj ifi>=j인관계도만족한다.

Recall:PrincipalComponentAnalysis(PCA)
SVDMethod
M
U VT
=
mxn mxr
rxn
Σ
rxr
min
P,Q
kP⌦(PQ) P⌦(M)k2
F = min
P,Q
kPQ Mk2
F
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- 만약에우리가행렬M의모든entry를알고있을때, 
rank가r인행렬M을가장잘설명하는rankk행렬을찾고싶다면?

M
U VT
=
mxn mxk
kxn
Σ
kxk
Recall:PrincipalComponentAnalysis(PCA)
SVDMethod
min
P,Q
kP⌦(PQ) P⌦(M)k2
F = min
P,Q
kPQ Mk2
F
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- 만약에우리가행렬M의모든entry를알고있을때, 
rank가r인행렬M을가장잘설명하는rankk행렬을찾고싶다면?
행렬M을SVD한이후에검은부분을날리면된다!
O(mn2
)<latexit sha1_base64="DPb5eAmsu0X12LAY8n0EU9wZF58=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXspuEdRb0Ys3K7i20K4lm2bb0CS7JFmhLP0RXjyoePX/ePPfmLZ70NYHA4/3ZpiZFyacaeO6305hZXVtfaO4Wdra3tndK+8fPOg4VYT6JOaxaodYU84k9Q0znLYTRbEIOW2Fo+up33qiSrNY3ptxQgOBB5JFjGBjpdZtVcjH+mmvXHFr7gxomXg5qUCOZq/81e3HJBVUGsKx1h3PTUyQYWUY4XRS6qaaJpiM8IB2LJVYUB1ks3Mn6MQqfRTFypY0aKb+nsiw0HosQtspsBnqRW8q/ud1UhNdBBmTSWqoJPNFUcqRidH0d9RnihLDx5Zgopi9FZEhVpgYm1DJhuAtvrxM/HrtsubdnVUaV3kaRTiCY6iCB+fQgBtogg8ERvAMr/DmJM6L8+58zFsLTj5zCH/gfP4AmgOOnw==</latexit><latexit sha1_base64="DPb5eAmsu0X12LAY8n0EU9wZF58=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXspuEdRb0Ys3K7i20K4lm2bb0CS7JFmhLP0RXjyoePX/ePPfmLZ70NYHA4/3ZpiZFyacaeO6305hZXVtfaO4Wdra3tndK+8fPOg4VYT6JOaxaodYU84k9Q0znLYTRbEIOW2Fo+up33qiSrNY3ptxQgOBB5JFjGBjpdZtVcjH+mmvXHFr7gxomXg5qUCOZq/81e3HJBVUGsKx1h3PTUyQYWUY4XRS6qaaJpiM8IB2LJVYUB1ks3Mn6MQqfRTFypY0aKb+nsiw0HosQtspsBnqRW8q/ud1UhNdBBmTSWqoJPNFUcqRidH0d9RnihLDx5Zgopi9FZEhVpgYm1DJhuAtvrxM/HrtsubdnVUaV3kaRTiCY6iCB+fQgBtogg8ERvAMr/DmJM6L8+58zFsLTj5zCH/gfP4AmgOOnw==</latexit><latexit sha1_base64="DPb5eAmsu0X12LAY8n0EU9wZF58=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXspuEdRb0Ys3K7i20K4lm2bb0CS7JFmhLP0RXjyoePX/ePPfmLZ70NYHA4/3ZpiZFyacaeO6305hZXVtfaO4Wdra3tndK+8fPOg4VYT6JOaxaodYU84k9Q0znLYTRbEIOW2Fo+up33qiSrNY3ptxQgOBB5JFjGBjpdZtVcjH+mmvXHFr7gxomXg5qUCOZq/81e3HJBVUGsKx1h3PTUyQYWUY4XRS6qaaJpiM8IB2LJVYUB1ks3Mn6MQqfRTFypY0aKb+nsiw0HosQtspsBnqRW8q/ud1UhNdBBmTSWqoJPNFUcqRidH0d9RnihLDx5Zgopi9FZEhVpgYm1DJhuAtvrxM/HrtsubdnVUaV3kaRTiCY6iCB+fQgBtogg8ERvAMr/DmJM6L8+58zFsLTj5zCH/gfP4AmgOOnw==</latexit><latexit sha1_base64="DPb5eAmsu0X12LAY8n0EU9wZF58=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXspuEdRb0Ys3K7i20K4lm2bb0CS7JFmhLP0RXjyoePX/ePPfmLZ70NYHA4/3ZpiZFyacaeO6305hZXVtfaO4Wdra3tndK+8fPOg4VYT6JOaxaodYU84k9Q0znLYTRbEIOW2Fo+up33qiSrNY3ptxQgOBB5JFjGBjpdZtVcjH+mmvXHFr7gxomXg5qUCOZq/81e3HJBVUGsKx1h3PTUyQYWUY4XRS6qaaJpiM8IB2LJVYUB1ks3Mn6MQqfRTFypY0aKb+nsiw0HosQtspsBnqRW8q/ud1UhNdBBmTSWqoJPNFUcqRidH0d9RnihLDx5Zgopi9FZEhVpgYm1DJhuAtvrxM/HrtsubdnVUaV3kaRTiCY6iCB+fQgBtogg8ERvAMr/DmJM6L8+58zFsLTj5zCH/gfP4AmgOOnw==</latexit>
Timecomplexity:

PrincipalComponentAnalysis(PCA)는사용할수없다.
SVDMethod
- 행렬M의모든entry를알때는PCA를사용할수있지만, 
행렬완성문제에서는행렬M의일부정보만알수있으므로PCA는사용할수없다.
- 행렬완성문제에서Globalminimum을구하는문제는NP-hard이다. 
그럼Localminimum이라도찾아야하지않을까?

SVDMethod
어떻게?AlternatingMinimization

SVDMethod
어떻게?AlternatingMinimization
NP-hard문제를풀때자주사용되는method중하나!EM도AM중하나!

SVDusingAlternatingMinimization
SVDMethod
M P Q≒ x
1. P를초기화한다.
2. P가고정인상태에서M을가장잘설명하는Q를구한다.
3. Q를고정시켜놓고,M을가장잘설명하는P를구한다.
4. 수렴할때까지반복한다.

1. P를초기화한다. ->예를들면,randominit.
2. P가고정인상태에서M을가장잘설명하는Q를구한다.->SVD로풀수있음(의사역행렬).
3. Q를고정시켜놓고,M을가장잘설명하는P를구한다.->SVD로풀수있음(의사역행렬).
SVDMethod
M P Q≒ x
질문!그냥P,Q를SGD로구하면왜안될까?

1. P를초기화한다. ->예를들면,randominit.
2. P가고정인상태에서M을가장잘설명하는Q를구한다.->SVD로풀수있음(의사역행렬).
3. Q를고정시켜놓고,M을가장잘설명하는P를구한다.->SVD로풀수있음(의사역행렬).
SVDMethod
M P Q≒ x
질문!그냥P,Q를SGD로구하면왜안될까?J.Besag(1986)에서OverfittingIssue가있음이보고!

Algorithm
SVDMethod
1: Input: observed set ⌦, values P⌦(M)
2: Initialize P0
randomly.
3: for t = 1, · · · , T:
4: Qt
arg min
Q2Rk⇥n
kP⌦(M Pt 1
Q)k2
F
5: Pt
arg min
P 2Rm⇥k
kP⌦(M PQt
)k2
F
6: Return ˆM = PT
QT
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M P Q≒ x

SVDMethod는LocalOptimum에빠질수밖에없다?
2013년에Jain등은[1]논문에서몇가지가정아래에서
SVDMethod는LocalOptimum에빠지지않는다는것을증명.
(물론알고리즘도살짝바꿨긴했다.)
SVDMethod

SVDMethod
Pt
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:t번째iteration에서의행렬P의값.ˆU == P, V †
== Q, V == QT
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SVDMethod
어렵게써있는것같지만,mini-batch를하라는뜻.

SVDMethod
앞서설명한AlternatingMinimization!

SVDMethod
초기화를SVD+Clipping을이용해서할때,
SVDmethod가찾는LocalOptimum은
GlobalOptimum에충분히가깝다!

SVDMethod
SrinadhBhojanapalli(2016NIPS)논문을통해
이초기화과정이필요없음이알려짐!
->Practical하게는RandomInitialPoint를설정하면됨.

SVDMethod
SrinadhBhojanapalli(2016NIPS)논문을통해
이초기화과정이필요없음이알려짐!
->Practical하게는RandomInitialPoint를설정하면됨.
*구현체:https://github.com/JoonyoungYi/MCAM-numpy

Low-rankMatrixCompletion의확률적접근.
• ProbabilisticMatrixFactorization(PMF)
• 2008년NIPS에발표된논문[2].
• 참고:NIPS에서는MatrixFactorization으로,ICML에서는MatrixCompletion으로부르는경향이있음.
ProbabilisticMatrixFactorization(PMF)

• MAP(MaximumAPosteriori)관점에서Low-rankMatrixCompletion을분석.
• 우리가관측한Rating이Low-rankStructure로이루어져있고 
관측과LatentVariable인P와Q에GaussianNoise가존재할때, 
가장그럴싸한(=가능성이높은,=확률이높은)P와Q를찾는것!

• OptimizationForm으로표현하면다음과같음.
• Given:
• hyperparameters:
max
P,Q
Pr[P, Q|M, ⌦, 2
, 2
P , 2
Q]
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M, ⌦, m, n<latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit><latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit><latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit><latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit>
, P , Q, k<latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit><latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit><latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit><latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit>

• OptimizationForm으로표현하면다음과같음.
• Given:
• hyperparameters:
max
P,Q
Pr[P, Q|M, ⌦, 2
, 2
P , 2
Q]
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사후확률,ObjectiveFunction
M, ⌦, m, n<latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit><latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit><latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit><latexit sha1_base64="o8bMUawi7BpolWHDSL6XcoONDQM=">AAAB93icbVBNSwMxEM36WetHqx69BIvgoZRdEdRb0YsXsYJrC+1SsulsG5pklyQr1KW/xIsHFa/+FW/+G9N2D9r6YODx3gwz88KEM21c99tZWl5ZXVsvbBQ3t7Z3SuXdvQcdp4qCT2Meq1ZINHAmwTfMcGglCogIOTTD4dXEbz6C0iyW92aUQCBIX7KIUWKs1C2Xbqq4cyugT6pYVLFVKm7NnQIvEi8nFZSj0S1/dXoxTQVIQznRuu25iQkyogyjHMbFTqohIXRI+tC2VBIBOsimh4/xkVV6OIqVLWnwVP09kRGh9UiEtlMQM9Dz3kT8z2unJjoPMiaT1ICks0VRyrGJ8SQF3GMKqOEjSwhVzN6K6YAoQo3NqmhD8OZfXiT+Se2i5t2dVuqXeRoFdIAO0THy0Bmqo2vUQD6iKEXP6BW9OU/Oi/PufMxal5x8Zh/9gfP5A8EDkWY=</latexit>
, P , Q, k<latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit><latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit><latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit><latexit sha1_base64="2F7ORcPToGchS07q5A+amCKnwoE=">AAACBnicbZDNSgMxFIXv1L9a/0ZdChIsgotSZkRQd0U3LltwbKEdhkyaaUMzmSHJCKV058ZXceNCxa3P4M63MW1H0NYLIR/n3EtyT5hyprTjfFmFpeWV1bXiemljc2t7x97du1NJJgn1SMIT2QqxopwJ6mmmOW2lkuI45LQZDq4nfvOeSsUScauHKfVj3BMsYgRrIwX2YUexXowraHYH9Qr6wUYFDQK77FSdaaFFcHMoQ171wP7sdBOSxVRowrFSbddJtT/CUjPC6bjUyRRNMRngHm0bFDimyh9N9xijY6N0UZRIc4RGU/X3xAjHSg3j0HTGWPfVvDcR//PamY4u/BETaaapILOHoowjnaBJKKjLJCWaDw1gIpn5KyJ9LDHRJrqSCcGdX3kRvNPqZdVtnJVrV3kaRTiAIzgBF86hBjdQBw8IPMATvMCr9Wg9W2/W+6y1YOUz+/CnrI9vIGeXwA==</latexit>

• 우리는이미다음과같은Low-rank관계를알고있음.
Pr[M|P, Q, 2
] =
nY
i=1
mY
j=1
[N[Mij|PiQj, 2
]]I⌦
ij ,
Pr[P| 2
P ] =
nY
i=1
N[Pi|0, 2
P I], Pr[Q| 2
Q] =
mY
j=1
N[Qi|0, 2
QI],
I⌦
ij =
⇢
1 (i, j) 2 ⌦
0 otherwise.<latexit sha1_base64="elL4BszW1wgEyU27bBjSnHWTNAE=">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</latexit><latexit sha1_base64="elL4BszW1wgEyU27bBjSnHWTNAE=">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</latexit><latexit sha1_base64="elL4BszW1wgEyU27bBjSnHWTNAE=">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</latexit><latexit sha1_base64="elL4BszW1wgEyU27bBjSnHWTNAE=">AAADlHicbZJbb9MwFIDThssot2488MDLERXTkKoqmZAYD0WDCYk+bCQSZZPqNHJct3XnXGQ7sMrNL+Lf8Ma/wWlT1LU7kqVz/c5FjjLOpHKcv7W6fe/+g4d7jxqPnzx99ry5f/BDprkgtE9SnoqrCEvKWUL7iilOrzJBcRxxehldn5Xxy59USJYm39U8o0GMJwkbM4KVcYX7td+HSNEbpT1RDM4XXhv8NiDJJjEeHgfQBZSJdDRMQs26blFZcahnpTUAFGM1JZjrC2Odm6RZAQvwQgZ+OINNUjDUvSH6FtMJXqYVJogaG829xSo59O5svNmo5C/AWePLilVCFOleERj/f6q/pvqb1PUCt6j+FtXfoZp5e8vhq0VKHqdjhXQDAEV0whKNhcDzQnNeGB+AC3AIR6wNs7eAWAJVoUGVUccEURylNzpVUyp+MUk7ZR2iyagiNZBgk6nqlCVhs+V0nKXAruJWSsuqxAubf9AoJXlME0U4lnLgOpkKDFkxwqlh55JmmFzjCR0YNcExlYFe/qoC3hjPCMapMC9RsPRuVmgcSzmPI5NZXklux0rnXbFBrsYngWZJliuakFWjcc5BpVB+URgxQYnic6NgIpiZFcgUC0yU+cjlEdztlXeV/nHnQ8f137VOP1bX2LNeWa+tI8u13lun1lfLs/oWqR/UT+qf6p/tl3bXPrO/rFLrtarmhXVL7It/+b4bmA==</latexit>

• 앞에서제시한objectivefunction(사후확률)에위식들을넣고로그취해전개하면,
ln Pr[P, Q|M, ⌦, 2
, 2
P , 2
Q]
=
1
2 2
mX
i=1
nX
j=1
I⌦
ij(Mij PiQj)2 1
2 2
P
nX
i=1
PT
i Pi
1
2 2
Q
mX
j=1
QjQT
j
1
2
((
nX
i=1
mX
j=1
I⌦
ij) ln 2
+ nk ln 2
P + mk ln 2
Q) + C
<latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit><latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit><latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">AAADcHicbVJdb9MwFHVTPkb56hgvCAEXKqqOrVUTIQEPQ5P2wh4mUmllk5omcly3cxs7UexMVCbP/D/e+BO88Adw2m402yxZPj7X595zrxwmEZOq2/1dsaq3bt+5u3Gvdv/Bw0eP65tPvsk4SwntkziK09MQSxoxQfuKqYieJinFPIzoSTg7KOIn5zSVLBbHap7QIccTwcaMYGWoYLPys+lFAjxFvyvtpvnA3YXej6Nd8L5yOsHmlGzCse9coMBdwz3fGYLn1faa0NbazsGLTTHQzqUMckPKjAea7dm5z1eXaXERcOgvy5joNIfW0fJsgxsw6AXTbd+ptUHbl2n/WyinFYXCP17obhL01gTTpQ+TvthGZPw311WOcQKtUvaS0uNYnYWhPsxL7mEbFpO8aHwHBMzWqMK1IXmJM8aMbgcOIKg3up3uYsF1YK9AA62WG9R/eaOYZJwKRSIs5cDuJmqocaoYiWhe8zJJE0xmeEIHBgrMqRzqxY/J4a1hRjCOU7OFggW7rtCYSznnoXlZdCuvxgryptggU+OPQ81EkikqyLLQOItAxVB8PxixlBIVzQ3AJGXGK5AznGKizCetmSHYV1u+DvpO51PH7r1v7H9eTWMDPUdvUAvZ6APaR1+Qi/qIVP5YT60X1kvrb/VZ9VX19fKpVVlptlBpVd/9AwA1DEY=</latexit><latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit>
Pr[M|P, Q, 2
] =
nY
i=1
mY
j=1
[N[Mij|PiQj, 2
]]I⌦
ij ,
Pr[P| 2
P ] =
nY
i=1
N[Pi|0, 2
P I], Pr[Q| 2
Q] =
mY
j=1
N[Qi|0, 2
QI],
I⌦
ij =
⇢
1 (i, j) 2 ⌦

• 앞에서제시한objectivefunction(사후확률)에위식들을넣고로그취해전개하면,
ln Pr[P, Q|M, ⌦, 2
, 2
P , 2
Q]
=
1
2 2
mX
i=1
nX
j=1
I⌦
ij(Mij PiQj)2 1
2 2
P
nX
i=1
PT
i Pi
1
2 2
Q
mX
j=1
QjQT
j
1
2
((
nX
i=1
mX
j=1
I⌦
ij) ln 2
+ nk ln 2
P + mk ln 2
Q) + C
<latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit><latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit><latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit><latexit sha1_base64="pRy3m8u483qCLKX8aDI1DZiFydI=">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</latexit>
정규분포식이expform이라log취하면식이편해짐.
Pr[M|P, Q, 2
] =
nY
i=1
mY
j=1
[N[Mij|PiQj, 2
]]I⌦
ij ,
Pr[P| 2
P ] =
nY
i=1
N[Pi|0, 2
P I], Pr[Q| 2
Q] =
mY
j=1
N[Qi|0, 2
QI],
I⌦
ij =
⇢
1 (i, j) 2 ⌦

Introduction to Low-rank Matrix Completion

Introduction to Low-rank Matrix Completion

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Introduction to Low-rank Matrix Completion