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Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization problem: Find the infimum over all semi-definite positive matrices $\Sigma$ of size $n\times n$ of the quantity: $$H = ||\phi(X^T \Sigma X)^{-1} Y||^2$$ where $\phi(.)$ is taken entrywise , $^T$ denotes the transpose operator, and $^{-1}$ denotes the inverse of the matrix.

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  • $\begingroup$ note that $X^\top\Sigma X$ need not be invertible. $\endgroup$ Sep 25, 2014 at 13:11
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    $\begingroup$ and of course if you do not say anything about $\phi$ then it looks pretty hopeless, $\endgroup$ Sep 25, 2014 at 13:13
  • $\begingroup$ Ok, if you want, assume that $\phi(x)=\tanh(x)$ if it helps. $\endgroup$
    – user16215
    Sep 26, 2014 at 13:00

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