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matrixfactorization.ppt
- 1. MAA704: Matrix Analysis Nonnegative Matrix Factorization Presented by: Filmon Tarik Tigabu
- 2. Matrix Factorization and its applications
- 3. Outline Expression power of matrix Various matrix factorization methods Application of matrix factorization
- 4. Curse of Dimensinality In machine learning and others, Dimension is a curse i.e., as they are intensive computation, and if dimension is high, this will worsen the computation, that’s why we have
- 5. Different matrix factorization methods LU decomposition Singular Value Decomposition(SVD) Probabilistic Matrix Factorization(PMF) Non-negative Matrix Factorization(NMF)
- 6. Application of matrix factorization LU decomposition ◦ Solving system of equations SVD decomposition ◦ Low rank matrix approximation ◦ Pseudo-inverse
- 7. Application of matrix factorization PMF ◦ Recommendation system NMF ◦ Learning the parts of objects
- 8. 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 ij R
- 9. 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?
- 10. 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.
- 11. PMF
- 12. PMF We want to maximize Equivalent to minimizing Can be solved using steepest descent method
- 13. Extension to PMF We can augment the model as long as we have additional data matrix that share comment latent feature matrix
- 14. 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
- 15. NMF How well can W and H approximate V How can we interpret the result
- 16. NMF Assumption ◦ ◦ ◦ Maximize logarithm likelihood and we get the objective function
- 17. 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”