Matrix Factorization and its applications

1 / 17

# Matrix Factorization and its applications - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Matrix Factorization and its applications

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Matrix Factorization and its applications

By Zachary

16th 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
• Distance matrix
Different matrix factorization methods
• LU decomposition
• Singular Value Decomposition(SVD)
• Probabilistic Matrix Factorization(PMF)
• Non-negative Matrix Factorization(NMF)
Application of matrix factorization
• LU decomposition
• Solving system of equations
• SVD decomposition
• Low rank matrix approximation
• Pseudo-inverse
Application of matrix factorization
• PMF
• Recommendation system
• NMF
• Learning the parts of objects
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
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?
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.
PMF
• We want to maximize
• Equivalent to minimizing
• Can be solved using steepest descent method
Extension to PMF
• We can augment the model as long as we have additional data matrix that share comment latent feature matrix
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
NMF
• How well can W and H approximate V
• How can we interpret the result
NMF
• Assumption
• Maximize logarithm likelihood and we get the objective function
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?
• Thank you