Matrix Factorization Techniques and Applications
Explore the concept of matrix factorization in data analysis, understanding the common factors between otakus and characters, and optimizing recommendations through gradient descent. Learn about latent factors, minimizing errors, and topic analysis using matrix factorization models like LSA and PLSA.
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Otakus v.s. No. of Figures A 5 3 0 1 B 4 3 0 1 C D 1 1 1 0 0 4 5 4 E 0 1 5 4 There are some common factors behind otakus and characters. http://www.quuxlabs.com/blog/2010/09/matrix-factorization-a-simple-tutorial-and- implementation-in-python/
The factors are latent. Otakus v.s. No. of Figures match A No one cares Not directly observable B C
?1 ?2 ?3 ?4 ?? ?? ?? ?? ?? A 5 3 0 1 B 4 3 0 1 Matrix X C 1 1 0 5 D 1 0 4 4 E 0 1 5 4 No. of Otakus = M No. of characters = N N No. of latent factor = K K N ?? ?1 5 ??1 ??2 ??1??2 rA r1 r2 K ?? ?1 4 M N rB ?? ?1 1 Singular value decomposition (SVD) Minimize Error Matrix X
?? ?1 ?2 ?3 ?4 ?? ?? ?? ?? ?? ?? ??1 A 5 3 ? 1 B 4 3 ? 1 C 1 1 ? 5 D 1 ? 4 4 E ? 1 5 4 Only considering the defined value Minimizing ?? ?1 5 ?? ?1 4 2 ?? ?? ??? ? = ?? ?1 1 ?,? Find ?? and ?? by gradient descent
?1 ?2 ?3 ?4 ?? ?? ?? ?? ?? -0.4 A 5 3 ? 1 -0.3 2.2 B 4 3 ? 1 C 1 1 ? 5 0.6 D 1 ? 4 4 0.1 E ? 1 5 4 Assume the dimensions of r are all 2 (there are two factors) A 0.2 2.1 1 ( 1 ( ) ) 0.0 2.2 B C D 0.2 1.3 1.9 1.8 0.7 0.2 2 ( 2 ( ) ) 3 ( 3 ( ) ) 0.1 1.5 1.9 -0.3 4 ( 4 ( ) ) 2.2 0.5 E 2.2 0.0
More about Matrix Factorization Considering the induvial characteristics ?? ?1+ ??+ ?1 5 ?? ?1 5 ??: otakus A likes to buy figures ?1: how popular character 1 is 2 ?? ??+ ??+ ?? ??? ? = Minimizing ?,? Find ??, ??, ??, ?? by gradient descent (can add regularization) Ref: Matrix Factorization Techniques For Recommender Systems
Matrix Factorization for Topic analysis character document, otakus word Latent semantic analysis (LSA) Number in Table: Doc 1 Doc 2 Doc 3 Doc 4 Term frequency (weighted by inverse document frequency) 5 3 0 1 4 0 0 1 1 1 0 5 Latent factors are topics ( ) 1 0 0 4 0 1 5 4 Probability latent semantic analysis (PLSA) Thomas Hofmann, Probabilistic Latent Semantic Indexing, SIGIR, 1999 latent Dirichlet allocation (LDA) David M. Blei, Andrew Y. Ng, Michael I. Jordan, Latent Dirichlet Allocation, Journal of Machine Learning Research, 2003
