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I'm trying to derive a result from this paper:

http://www.research.yahoo.net/files/HuKorenVolinsky-ICDM08.pdf

given cost function

[tex]

\min_{x_* , y_* } \sum_{u, i} c_{ui} (p_{ui} - x^T_u y_i)^2 + \lambda \left( \sum_u ||x_u||^2 + \sum_i ||y_||^2 \right)

[/tex]

Where both x_{u}and y_{i}are vectors in ℝ^{k}.

I want to find the minimum by using alternating least squares. Therefore I fix y and find the derivative with respect to x_{u}.

c_{ui}, p_{ui}and λ are constants.

They derive the following:

[tex]x_u = \left( Y^TC^uY + \lambda I \right) ^{-1} Y^T C^u p \left( u \right)[/tex]

I'm unsure how reproduce this result...

I don't know if I should try to derive with respect to the whole vector x_{u}or against one entry k (x_{uk}) in the vector and then try to map this to a function for the whole vector?

Any pointers are very much appreciated.

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# Derivation of least squares containing vectors

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