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

Abstract

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 over the nonnegative orthant of the unit sphere. After the magnitudes of the data are sorted, every canonical proximal direction belongs to a finite candidate set indexed by its support size \(k\in\{1,\ldots,n\}\). For each \(k\geq 2\), a candidate exists if and only if an explicit quartic equation has a simple root satisfying both a positive-orthant condition and a second-order derivative condition. This characterization eliminates the need to guess the unknown sparsity. We further derive a necessary-and-sufficient \(O(1)\) prefix test that enforces positivity and prunes support sizes for which no candidate can exist. Using prefix sums, the optimized method performs an \(O(n)\) scan after an \(O(n\log n)\) sorting step; the construction of multiple tied outputs is necessarily output-sensitive. Numerical experiments illustrate the finite-candidate selection, identify inputs on which a sparsity-guessing baseline is suboptimal, quantify the benefit of pruning, and demonstrate the numerical accuracy of the secular-equation solver. The observed runtimes exhibit near-linear growth over the tested range, consistent with the predicted \(O(n)\) post-sorting scan and \(O(n\log n)\) total complexity.