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Matrix Factorization and its applications

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yuval
17 diapositives · 764 Views · Aug 20, 2014

Matrix Factorization and its applications. By Zachary 16 th Nov, 2010. Outline. Expression power of matrix Various matrix factorization methods Application of matrix factorization. What can matrix represent?. System of equations User rating matrix Image Matrix structure in graph theory

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Matrix Factorization and its applications
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  1. Matrix Factorization and its applications By Zachary 16th Nov, 2010

  2. Outline • Expression power of matrix • Various matrix factorization methods • Application of matrix factorization

  3. What can matrix represent? • System of equations • User rating matrix • Image • Matrix structure in graph theory • Adjacent matrix • Distance matrix

  4. Different matrix factorization methods • LU decomposition • Singular Value Decomposition(SVD) • Probabilistic Matrix Factorization(PMF) • Non-negative Matrix Factorization(NMF)

  5. Application of matrix factorization • LU decomposition • Solving system of equations • SVD decomposition • Low rank matrix approximation • Pseudo-inverse

  6. Application of matrix factorization • PMF • Recommendation system • NMF • Learning the parts of objects

  7. PMF • Consider a typical recommendation problem • Given a n by m matrix R with some entries unknown • n rows represent n users • m columns represent m movies • Entry represent the ith user’s rating on the jth movie • We are interested in the unknown entries’ possible values • i.e. Predict users’ ratings

  8. PMF • We can model the problem as R=U’V • U (k by n) is the latent feature matrix for users • How much the user likes action movie? • How much the user likes comedy movie? • V (k by m) is the latent feature matrix for movies • To what extent is the movie an action movie? • To what extent is the movie a comedy movie?

  9. PMF • If we can learn U and V from existing ratings, then we can compute unknown entries by multiplying these two matrices. • Let’s consider a probabilistic approach.

  10. PMF

  11. PMF • We want to maximize • Equivalent to minimizing • Can be solved using steepest descent method

  12. Extension to PMF • We can augment the model as long as we have additional data matrix that share comment latent feature matrix

  13. NMF • Consider the following problem • M = 2429 facial images • Each image of size n = 19 by 19 = 361 • Matrix V = n by m is the original dataset • We want to approximate V by two lower rank matrix W (n by 49) and H (49 by m) • V ~ WH • Constraints • All entries of W and H are non-negative

  14. NMF • How well can W and H approximate V • How can we interpret the result

  15. NMF • Assumption • Maximize logarithm likelihood and we get the objective function

  16. Criticize of NMF • NMF doesn’t always give parts based result • Sparseness constraints • For more information, refer to “Non-negative matrix factorization with sparseness constrains”

  17. Questions? • Thank you