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For certain $\ell_1$-regularized optimization problems, a critical computational step is the soft threshold operator:

$\mathcal{S}_t(x) = \mathrm{sgn}(x)\circ \mathrm{max}(|x|-t)$

where $\circ$ is element-wise multiplication, or the Hadamard product. This operator is the solution to the minimization problem

$\arg\min_z t||z||_1+\frac{1}{2}||z-x||^2_2$

which makes it a kind of proximity operator. What I'd like to know is, is there a closed form solution for the proximity operator of $h(x) = ||xF||_1$ for arbitrary matrix $F$? That is, is there a closed form solution to

$\arg\min_z t||zF||_1 + \frac{1}{2}||z-x||^2_2$

? Thanks.

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