deletions | additions       diff --git a/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex b/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex  index 68dde88..cf77816 100644  --- a/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex  +++ b/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex 
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\newcommand{\reals}{\mathbb{R}}  \newcommand{\complex}{\mathbb{C}}  \newcommand{\Fu}{\mathcal{F}_u}  \newcommand{\nv}{\mathrn{n}_u}  \newcommand{\nv}{\mathrn{n}_u}  \paragraph{Sparse MRI Appendix A revisited}  Here we provide a detailed derivation of the 'complex' gradient of each term in the cost function  
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Df(m) = D\| \Fu m - y\|^2_{\ell_2} + \lambda D\| \Psi m\|_{\ell_1}.  \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 a composite function $h(\Fu m)$,   where $h : \complex^\nv \to \complex^\nk$defined as