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Introduction to Low-rank Matrix Completion
- 26. Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Low-rankMatrixCompletion(LRMC)
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>
- 27. Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Low-rankMatrixCompletion(LRMC)
Setofknownentries
그러니까기존데이터를가장잘설명하는Low-rankMatrix를찾는것!
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
<latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">AAACMXicbVDLSgMxFM34rPU16tJNsAgtaJkpgrorCqILcQpWhU4dMmnahiaZIckIZZxvcuOXCC50oeLWnzB9LLR6IHA45x5y7wljRpV2nBdranpmdm4+t5BfXFpeWbXX1q9UlEhM6jhikbwJkSKMClLXVDNyE0uCeMjIddg7HvjXd0QqGolL3Y9Jk6OOoG2KkTZSYJ/5nIog9XZgLYP+PfQ50l2MWOplQepfcNJBGSx6tRLc/dcrnpdMLDi5rQR2wSk7Q8C/xB2TAhjDC+wnvxXhhBOhMUNKNVwn1s0USU0xI1neTxSJEe6hDmkYKhAnqpkOT87gtlFasB1J84SGQ/VnIkVcqT4PzeRgazXpDcT/vEai2wfNlIo40UTg0UfthEEdwUF/sEUlwZr1DUFYUrMrxF0kEdam5bwpwZ08+S+pV8qHZbe2V6gejdvIgU2wBYrABfugCk6BB+oAgwfwDN7Au/VovVof1udodMoaZzbAL1hf37ryqNM=</latexit><latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">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</latexit><latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">AAACMXicbVDLSgMxFM34rPU16tJNsAgtaJkpgrorCqILcQpWhU4dMmnahiaZIckIZZxvcuOXCC50oeLWnzB9LLR6IHA45x5y7wljRpV2nBdranpmdm4+t5BfXFpeWbXX1q9UlEhM6jhikbwJkSKMClLXVDNyE0uCeMjIddg7HvjXd0QqGolL3Y9Jk6OOoG2KkTZSYJ/5nIog9XZgLYP+PfQ50l2MWOplQepfcNJBGSx6tRLc/dcrnpdMLDi5rQR2wSk7Q8C/xB2TAhjDC+wnvxXhhBOhMUNKNVwn1s0USU0xI1neTxSJEe6hDmkYKhAnqpkOT87gtlFasB1J84SGQ/VnIkVcqT4PzeRgazXpDcT/vEai2wfNlIo40UTg0UfthEEdwUF/sEUlwZr1DUFYUrMrxF0kEdam5bwpwZ08+S+pV8qHZbe2V6gejdvIgU2wBYrABfugCk6BB+oAgwfwDN7Au/VovVof1udodMoaZzbAL1hf37ryqNM=</latexit><latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">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</latexit>
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>
- 28. Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Low-rankMatrixCompletion(LRMC)
Setofknownentries
그러니까기존데이터를가장잘설명하는Low-rankMatrix를찾는것!
아쉽게도이문제는NP-Hard!
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 ⌦,
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>
- 29. Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Low-rankMatrixCompletion(LRMC)
Setofknownentries
그러니까기존데이터를가장잘설명하는Low-rankMatrix를찾는것!
아쉽게도이문제는NP-Hard!
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
<latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">AAACMXicbVDLSgMxFM34rPU16tJNsAgtaJkpgrorCqILcQpWhU4dMmnahiaZIckIZZxvcuOXCC50oeLWnzB9LLR6IHA45x5y7wljRpV2nBdranpmdm4+t5BfXFpeWbXX1q9UlEhM6jhikbwJkSKMClLXVDNyE0uCeMjIddg7HvjXd0QqGolL3Y9Jk6OOoG2KkTZSYJ/5nIog9XZgLYP+PfQ50l2MWOplQepfcNJBGSx6tRLc/dcrnpdMLDi5rQR2wSk7Q8C/xB2TAhjDC+wnvxXhhBOhMUNKNVwn1s0USU0xI1neTxSJEe6hDmkYKhAnqpkOT87gtlFasB1J84SGQ/VnIkVcqT4PzeRgazXpDcT/vEai2wfNlIo40UTg0UfthEEdwUF/sEUlwZr1DUFYUrMrxF0kEdam5bwpwZ08+S+pV8qHZbe2V6gejdvIgU2wBYrABfugCk6BB+oAgwfwDN7Au/VovVof1udodMoaZzbAL1hf37ryqNM=</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=">AAACMXicbVDLSgMxFM34rPU16tJNsAgtaJkpgrorCqILcQpWhU4dMmnahiaZIckIZZxvcuOXCC50oeLWnzB9LLR6IHA45x5y7wljRpV2nBdranpmdm4+t5BfXFpeWbXX1q9UlEhM6jhikbwJkSKMClLXVDNyE0uCeMjIddg7HvjXd0QqGolL3Y9Jk6OOoG2KkTZSYJ/5nIog9XZgLYP+PfQ50l2MWOplQepfcNJBGSx6tRLc/dcrnpdMLDi5rQR2wSk7Q8C/xB2TAhjDC+wnvxXhhBOhMUNKNVwn1s0USU0xI1neTxSJEe6hDmkYKhAnqpkOT87gtlFasB1J84SGQ/VnIkVcqT4PzeRgazXpDcT/vEai2wfNlIo40UTg0UfthEEdwUF/sEUlwZr1DUFYUrMrxF0kEdam5bwpwZ08+S+pV8qHZbe2V6gejdvIgU2wBYrABfugCk6BB+oAgwfwDN7Au/VovVof1udodMoaZzbAL1hf37ryqNM=</latexit>
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=">AAACgXicbVFdb9MwFHXCgFG+Onjk5YpqaENTlkxIbEJIE7zwRtEom1RXkePepN4cJ7KdbZWVH8Lf4o1fA04bIdi4kqWjc889ts/NaimMjeOfQXhn4+69+5sPBg8fPX7ydLj17JupGs1xwitZ6bOMGZRC4cQKK/Gs1sjKTOJpdvGx659eojaiUl/tssZZyQolcsGZ9VQ6/E5LZhecSTduU0c/l1iwFnZOduE9UIm5pW4AQDMshHJMa7ZsnZSt5wBOUifOW4BXQMusunYiBz8q9uB8F6hQsHbbA0pX8viPsLIL1FfCYNQZUVTz3npAtSgWNupG0uEojuJVwW2Q9GBE+hqnwx90XvGmRGW5ZMZMk7i2M+9sBZfovRuDNeMXrMCph4qVaGZuFWEL256ZQ15pf5SFFfv3hGOlMcsy88ouMHOz15H/600bmx/OnFB1Y1Hx9UV5I8FW0O0D5kIjt3LpAeNa+LcCXzDNuPVb60JIbn75NpgcREdR8uXN6PhDn8YmeUFekh2SkLfkmHwiYzIhnPwKtoMo2A83wtdhHB6spWHQzzwn/1T47jewu7zY</latexit><latexit sha1_base64="DQjr8Uh3EhviIVyXrCXgxDxtHgk=">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</latexit>
아직은polynomialtime안에해를찾을수없다!
- 30. Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Low-rankMatrixCompletion(LRMC)
Setofknownentries
그러니까기존데이터를가장잘설명하는Low-rankMatrix를찾는것!
아쉽게도이문제는NP-Hard!
그러면,어떻게풀까?
min
P,Q
kP⌦(PQ) P⌦(M)k2
F
<latexit sha1_base64="8Odjzm3wkLPO/SU2OJ5vn1RB1Q8=">AAACMXicbVDLSgMxFM34rPU16tJNsAgtaJkpgrorCqILcQpWhU4dMmnahiaZIckIZZxvcuOXCC50oeLWnzB9LLR6IHA45x5y7wljRpV2nBdranpmdm4+t5BfXFpeWbXX1q9UlEhM6jhikbwJkSKMClLXVDNyE0uCeMjIddg7HvjXd0QqGolL3Y9Jk6OOoG2KkTZSYJ/5nIog9XZgLYP+PfQ50l2MWOplQepfcNJBGSx6tRLc/dcrnpdMLDi5rQR2wSk7Q8C/xB2TAhjDC+wnvxXhhBOhMUNKNVwn1s0USU0xI1neTxSJEe6hDmkYKhAnqpkOT87gtlFasB1J84SGQ/VnIkVcqT4PzeRgazXpDcT/vEai2wfNlIo40UTg0UfthEEdwUF/sEUlwZr1DUFYUrMrxF0kEdam5bwpwZ08+S+pV8qHZbe2V6gejdvIgU2wBYrABfugCk6BB+oAgwfwDN7Au/VovVof1udodMoaZzbAL1hf37ryqNM=</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=">AAACMXicbVDLSgMxFM34rPU16tJNsAgtaJkpgrorCqILcQpWhU4dMmnahiaZIckIZZxvcuOXCC50oeLWnzB9LLR6IHA45x5y7wljRpV2nBdranpmdm4+t5BfXFpeWbXX1q9UlEhM6jhikbwJkSKMClLXVDNyE0uCeMjIddg7HvjXd0QqGolL3Y9Jk6OOoG2KkTZSYJ/5nIog9XZgLYP+PfQ50l2MWOplQepfcNJBGSx6tRLc/dcrnpdMLDi5rQR2wSk7Q8C/xB2TAhjDC+wnvxXhhBOhMUNKNVwn1s0USU0xI1neTxSJEe6hDmkYKhAnqpkOT87gtlFasB1J84SGQ/VnIkVcqT4PzeRgazXpDcT/vEai2wfNlIo40UTg0UfthEEdwUF/sEUlwZr1DUFYUrMrxF0kEdam5bwpwZ08+S+pV8qHZbe2V6gejdvIgU2wBYrABfugCk6BB+oAgwfwDN7Au/VovVof1udodMoaZzbAL1hf37ryqNM=</latexit>
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>
- 31. Low-rankMatrixCompletion의수학적정의
-먼저PΩ(S)를정의하자.
Low-rankMatrixCompletion(LRMC)
Setofknownentries
그러니까기존데이터를가장잘설명하는Low-rankMatrix를찾는것!
아쉽게도이문제는NP-Hard!
그러면,어떻게풀까?
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 ⌦,
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>
SVDMethod.
- 32. 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!
- 33. 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.
- 34. 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?
- 35. 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:
- 40. 1. P를초기화한다. ->예를들면,randominit.
2. P가고정인상태에서M을가장잘설명하는Q를구한다.->SVD로풀수있음(의사역행렬).
3. Q를고정시켜놓고,M을가장잘설명하는P를구한다.->SVD로풀수있음(의사역행렬).
4. 수렴할때까지반복한다.
SVDusingAlternatingMinimization
SVDMethod
M P Q≒ x
질문!그냥P,Q를SGD로구하면왜안될까?
- 41. 1. P를초기화한다. ->예를들면,randominit.
2. P가고정인상태에서M을가장잘설명하는Q를구한다.->SVD로풀수있음(의사역행렬).
3. Q를고정시켜놓고,M을가장잘설명하는P를구한다.->SVD로풀수있음(의사역행렬).
4. 수렴할때까지반복한다.
SVDusingAlternatingMinimization
SVDMethod
M P Q≒ x
질문!그냥P,Q를SGD로구하면왜안될까?J.Besag(1986)에서OverfittingIssue가있음이보고!
- 42. 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
- 52. Low-rankMatrixCompletion의확률적접근.
ProbabilisticMatrixFactorization(PMF)
• ProbabilisticMatrixFactorization(PMF)
• 2008년NIPS에발표된논문[2].
• 참고:NIPS에서는MatrixFactorization으로,ICML에서는MatrixCompletion으로부르는경향이있음.
• 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>
- 53. Low-rankMatrixCompletion의확률적접근.
ProbabilisticMatrixFactorization(PMF)
• ProbabilisticMatrixFactorization(PMF)
• 2008년NIPS에발표된논문[2].
• 참고:NIPS에서는MatrixFactorization으로,ICML에서는MatrixCompletion으로부르는경향이있음.
• 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]
<latexit sha1_base64="vpJODlatchobO02oPy7INUnRjHM=">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</latexit><latexit sha1_base64="vpJODlatchobO02oPy7INUnRjHM=">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</latexit><latexit sha1_base64="vpJODlatchobO02oPy7INUnRjHM=">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</latexit><latexit sha1_base64="vpJODlatchobO02oPy7INUnRjHM=">AAACL3icbZBNSxxBEIZ7NInr5mvVYy6NSyAHWWYWIXoTPZiLOAtZFXbGoaa3dm3snhm6a8Rl3J/kJT9FLwpGcs2/SO9HwGgKmn76rSqq600LJS35/r23sPjq9Zul2nL97bv3Hz42VlaPbF4agV2Rq9ycpGBRyQy7JEnhSWEQdKrwOD3fm+SPL9BYmWffaVRgrGGYyYEUQE5KGvuRhsukCjd4Z8wjwkuqQjPuTd5XBxs8OtQ4BHdbOdRw2v5LSfiEO6ftOGk0/ZY/Df4Sgjk02TzCpHET9XNRasxIKLC2F/gFxRUYkkLhuB6VFgsQ5zDEnsMMNNq4mi485p+d0ueD3LiTEZ+qTzsq0NaOdOoqNdCZfZ6biP/L9UoabMWVzIqSMBOzQYNSccr5xD3elwYFqZEDEEa6v3JxBgYEOY/rzoTg+covodtubbeCzmZzZ3fuRo19YuvsCwvYV7bDvrGQdZlg1+yWPbCf3g/vznv0fs1KF7x5zxr7J7zffwAQIadf</latexit>
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>
- 54. ProbabilisticMatrixFactorization(PMF)
• 우리는이미다음과같은Low-rank관계를알고있음.
ProbabilisticMatrixFactorization(PMF)
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=">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</latexit>
- 55. ProbabilisticMatrixFactorization(PMF)
• 우리는이미다음과같은Low-rank관계를알고있음.
• 앞에서제시한objectivefunction(사후확률)에위식들을넣고로그취해전개하면,
ProbabilisticMatrixFactorization(PMF)
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
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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>
- 56. ProbabilisticMatrixFactorization(PMF)
• 우리는이미다음과같은Low-rank관계를알고있음.
• 앞에서제시한objectivefunction(사후확률)에위식들을넣고로그취해전개하면,
ProbabilisticMatrixFactorization(PMF)
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=">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><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 ⌦
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