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Proximal operator or mapping

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What is the proximal operator?

The proximal operator is used to make an approximation to a value, while making a compromise between the accuracy of the approximation and a cost associated to the new value. Say we have the following minimization problem: $$\underset{\mathbf{u}}{\min } \frac{1}{2}\left\Vert \mathbf{u} - \mathbf{x}\right\Vert _2^2.$$ It is clear that the best approximation to $\mathbf{x}$ is the $\mathbf{x}$ itself ($\mathbf{u} = \mathbf{x}$). Now we add a cost to the choice we make of $\mathbf{u}$: $$\textrm{prox}_h(\mathbf{x}) = \underset{\mathbf{u}}{\textrm{argmin }} \frac{1}{2}\left\Vert \mathbf{u} - \mathbf{x}\right\Vert _2^2 + h(\mathbf{u}).$$ Now the best approximation depends how costly it is to choose $\mathbf{u} = \mathbf{x}$, given $h(\mathbf{u})$, and a compromise must be made. The first example is the case that $h(\mathbf{u}) = 0$. The proximal operator $\textrm{prox}_h(\mathbf{x})$ is the analytic solution to the previous minimization problem.


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