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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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Presentation Transcript
outline
Outline
  • Expression power of matrix
  • Various matrix factorization methods
  • Application of matrix factorization
what can matrix represent
What can matrix represent?
  • System of equations
  • User rating matrix
  • Image
  • Matrix structure in graph theory
    • Adjacent matrix
    • Distance matrix
different matrix factorization methods
Different matrix factorization methods
  • LU decomposition
  • Singular Value Decomposition(SVD)
  • Probabilistic Matrix Factorization(PMF)
  • Non-negative Matrix Factorization(NMF)
application of matrix factorization
Application of matrix factorization
  • LU decomposition
    • Solving system of equations
  • SVD decomposition
    • Low rank matrix approximation
    • Pseudo-inverse
application of matrix factorization6
Application of matrix factorization
  • PMF
    • Recommendation system
  • NMF
    • Learning the parts of objects
slide7
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
slide8
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?
slide9
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.
slide11
PMF
  • We want to maximize
  • Equivalent to minimizing
  • Can be solved using steepest descent method
extension to pmf
Extension to PMF
  • We can augment the model as long as we have additional data matrix that share comment latent feature matrix
slide13
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
slide14
NMF
  • How well can W and H approximate V
  • How can we interpret the result
slide15
NMF
  • Assumption
    • Maximize logarithm likelihood and we get the objective function
criticize of nmf
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”
questions
Questions?
  • Thank you