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Proximity Operator of the $\ell_1$ over $\ell_2$ Function
We study the complete, possibly set-valued proximity operator of the nonconvex, scale-invariant ratio \(h(\vx)=\|\vx\|_{1}/\|\vx\|_{2}\). A polar decomposition of the nonzero branch, followed by sign and permutation reductions, transforms the proximal problem into a smooth rank-one quadratic problem
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