Proximity Operator of the $\ell_1$ over $\ell_2$ Function
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| Format: | Preprint |
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2026
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| author | Shen, Lixin Song, Guohui |
| author_facet | Shen, Lixin Song, Guohui |
| contents | We study the proximity operator of the nonconvex, scale-invariant ratio $h(\vx)=\|\vx\|_{1}/\|\vx\|_{2}$ and show it can be computed exactly in any dimension. By expressing $\vx=r\vu$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16128 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proximity Operator of the $\ell_1$ over $\ell_2$ Function Shen, Lixin Song, Guohui Optimization and Control We study the proximity operator of the nonconvex, scale-invariant ratio $h(\vx)=\|\vx\|_{1}/\|\vx\|_{2}$ and show it can be computed exactly in any dimension. By expressing $\vx=r\vu$ 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. |
| title | Proximity Operator of the $\ell_1$ over $\ell_2$ Function |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2601.16128 |