deletions | additions       diff --git a/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex b/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex  index e2a8edf..917d30d 100644  --- a/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex  +++ b/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex 
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\newcommand{\nk}{n_k}  \subsubsection*{Sparse MRI Appendix A revisited}  Here we provide a detailed derivation of the 'complex' gradient of each term in the cost function as defined in Equation [A1]. Since by definition the 'complex' gradient is the transpose of the 'complex' derivative, we first find expressions in terms of derivatives. The cost function $f : \complex^{\nv} \to \reals$ is given by  \begin{equation}  f(m) = \| \Fu m - y\|^2_{\ell_2} + \lambda \| \Psi m\|_{\ell_1}. 
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\end{equation}  Let's look at each term separately. For the first term, $D\| \Fu m - y\|^2_{\ell_2}$,   we apply the chain rule to the composite function $h(\Fu m - y)$,  where $h(x) = \|x\|^2_{\ell_2} : \complex^\nv \complex^{\nv}  \to \complex^\nk$. \complex^{\nk}$.  It follows that  \begin{eqnarray}  D\| \Fu m - y\|^2_{\ell_2} &=& Dh(\Fu m - y) D(\Fu m - y) \nonumber\\  &=& 2(\Fu m - y)^T \Fu. \nonumber  \end{eqnarray}  Lustig et al. used a smooth approximation of the absolute value of a complex number $x = a + jb \in \complex$, given as  \begin{equation}  |x| \approx \sqrt{x^*x + \mu}, \nonumber 
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applying the definition of the complex derivative yields  \begin{eqnarray}  D|x| &=& \frac{d|x|}{da} + j\frac{d|x|}{db} \nonumber\\  &\approx& \frac{d}{da} \left( \sqrt{(a + jb)^*(a + jb) + \mu} \right) + j\frac{d}{db} \left( \sqrt{(a + jb)^*(a + jb) + \mu} \right) \nonumber\\ &\approx& \frac{d}{da} \left( \sqrt{a^2 + b^2 + \mu} \right) + j\frac{d}{db} \left( \sqrt{a^2 + b^2 + \mu} \right) \nonumber\\ &\approx& \frac{2a}{2\sqrt{a^2 + b^2 + \mu}} + j\frac{2b}{2\sqrt{a^2 + b^2 + \mu}} \nonumber\\ &\approx& \frac{x}{\sqrt{x^*x + \mu}}. \nonumber \end{eqnarray}  For the second term, $D\| \Psi m \|_{\ell_1}$,   we apply the chain rule to the composite function $h = g \circ f : \complex^{\nv} \to \reals$ given by  \begin{equation}  $g(x) = \|x\|_{\ell_1} : \complex^{\nk} \to \reals$.  $f(m) = \Fu m - y : \complex^{\nv} \to \complex^\nk$.  \end{equation}  It follows that  \begin{eqnarray}  D\| \Fu m - y\|^2_{\ell_2} &=& Dh(\Fu m - y) D(\Fu m - y) \nonumber\\  &=& 2(\Fu m - y)^T \Fu. \nonumber  \end{eqnarray}