Proximity Operator of the ${\mathrm{\ell }}_1$ over ${\mathrm{\ell }}_2$ Function

We study the proximity operator of the nonconvex, scale-invariant ratio $h(\text{vx})=\mathrm{\parallel }\text{vx}{\mathrm{\parallel }}_1\mathrm{/}\mathrm{\parallel }\text{vx}{\mathrm{\parallel }}_2$ and show it can be computed exactly in any dimension. By expressing $\text{vx}=r\text{vu}$ and exploiting sign and permutation invariance, we reduce the proximal step to a smooth optimization of a rank-one quadratic over the nonnegative orthant of the unit sphere. We prove that every proximal point arises from a finite candidate set indexed by $k\in \{1,\dots ,n\}$: the active subvector is a local, but nonglobal, minimizer on ${\mathbb{S}}^{k-1}$ characterized by the roots of an explicit quartic. This yields closed-form candidates, an exact selection rule, and a necessary and sufficient existence test. Building on these characterizations, we develop practical algorithms, including an $O(n)$ implementation via prefix sums and a pruning criterion that avoids unnecessary quartic solves. The method returns all proximal points when the prox is non-unique, and in experiments it attains strictly lower objective values than approaches that guess sparsity or rely on sphere projections with limited scalability.