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Im trying to solve this equation with respect to $m$ which is a $K×1$ vector:

$$\boldsymbol{\Sigma}^{-1}\boldsymbol{m}+\boldsymbol{W}^{T}\frac{\boldsymbol{1}-\exp(\boldsymbol{Wm}+\boldsymbol{W_0})}{\boldsymbol{1}+\exp(\boldsymbol{Wm}+\boldsymbol{W_0})}=\boldsymbol{W}^{T}(\boldsymbol{y}-\textbf{1/2})$$

where $\boldsymbol{\Sigma}$ is a $K×K$ matrix, $\textbf{W}$ a $D×K$ matrix, $\textbf{W}_0$ $D×1$ vector and $\textbf{y}$ a $K×1$ vector.

How can I handle the denominator in the fraction? Is there any general methodology for equations of this type?

Paul Sinclair
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tata
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    What do you mean by exp of a vector? – Paul Mar 22 '21 at 19:01
  • @Paul the exponential function applied elementwise – tata Mar 22 '21 at 19:06
  • What is a ratio of two vectors then, also element wise? – Paul Mar 22 '21 at 19:35
  • @Paul Yes, exactly. – tata Mar 22 '21 at 19:47
  • You are trying to pretend this is a linear system, but it is not. I suggest treating it as $K$ equations in $K$ separate unknowns. The fraction can be replaced with $\tanh\left(\frac{Wm + W_0}2\right)$, but that is just window-dressing. The general procedure is to solve for one variable in terms of the others, substitute, then solve the new system of $K-1$ variables in $K-1$ unknowns. Repeat until solvable.Because $m$ is inside and outside the exp, there is no nice solution. In real life, you'll have to solve numerically. – Paul Sinclair Mar 23 '21 at 02:53

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