Proximity Operator of the $\ell_1$ over $\ell_2$ Function

TL;DR

This paper derives an exact, scalable method to compute the proximity operator of a nonconvex ratio function involving $ ext{l}_1$ and $ ext{l}_2$ norms, enabling improved optimization in high dimensions.

Contribution

It provides a closed-form, exact algorithm for the proximity operator of the nonconvex ratio function, including practical $O(n)$ implementation and a comprehensive characterization of solutions.

Findings

Exact computation of the proximity operator in any dimension.

The proposed method outperforms existing approaches in objective value.

Efficient $O(n)$ algorithm with pruning for large-scale problems.

Abstract

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.