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I'm studying Matrix Factorization (to use in Recommender Systems as link predictor) and i want to know if there is any similarity with PCA? The latent features can be compared to the eigenvectors?

Thank you

kjetil b halvorsen
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Augusto
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    I believe there are many forms of matrix factorization. What one do you mean? I'd also ask you to unpack your question, make it more specific. As far as you're studying the subject you might have faced concrete issues. – ttnphns Nov 11 '12 at 08:35
  • Well... At this moment, i am trying to understand this algorithm: http://www.quuxlabs.com/blog/2010/09/matrix-factorization-a-simple-tutorial-and-implementation-in-python/ – Augusto Nov 11 '12 at 16:59
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    Augusto, beware of relying on material that does not distinguish one kind of matrix factorization from another. Although it *might* be correct, it belies a lack of understanding on the part of the author(s). This is a huge red flag, because numerical algorithms with matrices can be notoriously difficult to get right: they can look ok but become unstable with large matrices, or fail for certain special kinds of matrices, etc. Anyone "rolling their own code" had better know what they're doing. A far better resource is the [Numerical Recipes pages](http://www.nr.com/). – whuber Nov 12 '12 at 20:46

3 Answers3

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In a sense, PCA is a kind of matrix factorization, since it decomposes a matrix $\bf{X}$ into $\bf{W{\Sigma}V^T}$. However, matrix factorization is a very general term.

Also, see this answer on math.stackexchange.

Community
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January
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PCA, as far as I can tell, is just looking at and doing stuff with the output of factorisation through the Singular Value Decomposition. $\bf{X=U \Sigma V^T}$.

Just in case the question was about factorisation generally. Factorisation would be any breaking up / decomposing of a matrix into a product of other matrices. A different factorisation is the LU, $\bf{X=LU}$, which helps solve systems of equations through elimination.

conjectures
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There are indeed many Matrix Factorization techniques. You might be interested in this page: https://sites.google.com/site/igorcarron2/matrixfactorizations and thyen let us know which matrix factorization you want to compare with PCA.

Igor Carron
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