What is a Key Matrix?

What is a good way do svd matrix factorization for recommendation systems?

• Hi,  I am trying to learn latent models in recommendation systems which is nothing but factorization of a user-item matrix. I was going thru this slides [http://www.hpi.uni-potsdam.de/fileadmin/hpi/FG_Naumann/lehre/SS2011/Collaborative_Filtering/pres1-matrixfactorization.pdf%5D%5B1] In slide 6: They decomposed a matrix M (user-item matrix) into = **user feature matrix *****movie feature matrix** And they say they did SVD  but isnt SVD suppose to give three matrices U,S,V? If I plug the values in octave, I get this as SVD decomposition of that matrix: > u = > >   -0.825066  -0.047735  -0.563016  > -0.443084  -0.563669   0.697104    -0.350631   0.824620   0.443914 > > s = > > Diagonal Matrix > >    9.2654        0        0 >         0   3.2340        0 >         0        0   1.3016 > > v = > >   -0.636527  -0.770982  -0.020485  > -0.476317   0.372083   0.796666     -0.606593   0.516857  -0.604073 How did the author got those values? :( Please suggest Thanks   [1]: http://www.hpi.uni-potsdam.de/fileadmin/hpi/FG_Naumann/lehre/SS2011/Collaborative_Filtering/pres1-matrixfactorization.pdf

• Answer:

  The SVD does indeed produce three matrices. U and V's columns are orthonormal bases for the user-feature and movie-feature space respectively. The process has also normalized out the singular values, kind of like scaling factors, into the matrix S. Usually you zero out the long tail of small values in S and drop the corresponding columns in U and V to get much smaller, slightly approximate factorization. That's part of what you need to do with S. When writing the factorization you'll sometimes see people push S into U and V and show it as the product of two matrices. You can put S with U, or V, or split it by putting sqrt(S) (a diagonal matrix holding the square roots of the values in S). Compare those possibilities to the results in the paper. If you're not necessarily only interested in the SVD, there are other ways to make a low-rank approximate factorization, that don't go so far as to extract the singular values (and are simpler as a result). Look up nonnegative matrix factorization. Alternating Least Squares is an easy and effective approach, for example. It's implemented in Mahout (http://mahout.apache.org) and Myrrix (http://myrrix.com). Forms of ALS can be better for recommendation, depending on your input.