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Here is an optimization problem I came across in the paper titled "Learning the dependency structure of latent factors":

And here is the closed-form solution of it for S:

enter image description here

where

X is a PxN data matrix where P is the number of variables and N is the number of samples,

B is a PxK basis matrix which projects data X to a K-dimensional basis,

S is a KxN latent factor matrix of K factors,

Phi is a KxK symmetric matrix containing the structure among factors,

and sigma and rho are the scalars.

I didn't understand how the authors get to this derivation, and will be happy if someone can explain it. Thanks.

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    $\begingroup$ You need to provide much more information than this. What are $B$, $S$ and $\Phi$? What is $N$, $K$, $X$, $\sigma^2$, $\rho$? $\endgroup$ – Greenparker May 10 '16 at 2:41
  • $\begingroup$ Equations (12), (13), and Algorithm 1 of papers.nips.cc/paper/… , which references cs.utexas.edu/~pradeepr/paperz/invcov.pdf . That solution S "For subproblem of S we have closed-form solution ..." - I don't know what subproblem that corresponds to, but it is used iteratively as part of Algorithm 1, not as an overall solution to the original problem, and notice that if doesn't tell you the optimal values of those variables on the RHS, which is why it's only part of a larger iterative process. $\endgroup$ – Mark L. Stone May 10 '16 at 2:54
  • $\begingroup$ Ultimately, the authors might be the only ones who can explain it. $\endgroup$ – Mark L. Stone May 10 '16 at 2:56
  • $\begingroup$ Yes, I know that it is an iterative process. They optimize this big function for three different parameters (S, B and Phi). I have problem only with understanding minimizing for S. The others are more intuitive. $\endgroup$ – user5054 May 10 '16 at 2:57
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    $\begingroup$ Updated the question so that it includes details about the variables in the formula. $\endgroup$ – user5054 May 10 '16 at 3:08

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