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

Lixin Shen, Guohui Song

TL;DR

The paper addresses computing the proximity operator of the nonconvex, scale-invariant ratio $h(\mathbf{x})=\|\mathbf{x}\|_1/\|\mathbf{x}\|_2$. It recasts the problem as a smooth optimization on the unit sphere and proves that all proximal points originate from a finite set of local non-global minimizers characterized by the roots of an explicit quartic, enabling exact computation. It then develops both a naive $O(n^2)$ and an optimized $O(n)$ algorithm based on prefix sums to enumerate candidate active sets and recover the proximal points, including all proximal points when non-unique. Numerical experiments show exact solutions and strictly better objective values than baselines that rely on sparsity guessing or low-dimensional sphere projections, demonstrating improved accuracy and scalability in high dimensions.

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.

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

TL;DR

The paper addresses computing the proximity operator of the nonconvex, scale-invariant ratio . It recasts the problem as a smooth optimization on the unit sphere and proves that all proximal points originate from a finite set of local non-global minimizers characterized by the roots of an explicit quartic, enabling exact computation. It then develops both a naive and an optimized algorithm based on prefix sums to enumerate candidate active sets and recover the proximal points, including all proximal points when non-unique. Numerical experiments show exact solutions and strictly better objective values than baselines that rely on sparsity guessing or low-dimensional sphere projections, demonstrating improved accuracy and scalability in high dimensions.

Abstract

We study the proximity operator of the nonconvex, scale-invariant ratio and show it can be computed exactly in any dimension. By expressing 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 : the active subvector is a local, but nonglobal, minimizer on 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 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.
Paper Structure (7 sections, 4 theorems, 45 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 7 sections, 4 theorems, 45 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proximity operator via manifold optimization
  3. Structural characterizations of the proximity operator
  4. Algorithm to compute the exact proximity operator
  5. Further improvements
  6. Numerical experiments
  7. Conclusion

Key Result

Lemma 1

The minimizer $\bm{u}_{\frak{\bm{y}}}$ of the function $F_{\frak{\bm{y}}}$ in eq:vu-abs(vy) belongs to $\mathbb{R}_{{\hbox{$\bm{\downarrow}$}}}^{n}$.

Figures (4)

  • Figure 1: The values of $A_{k}$\ref{['eq:A-k']} for $2\leq k\leq 8$.
  • Figure 2: The function values $F_{\frak{\bm{y}}}(\overline{\bm{u}}_{k})$ for $1\leq k\leq 9$.
  • Figure 3: Function values $Q_{\bm{y}}(\bm{x}_{k})$ in \ref{['eg:comparison-2']}.
  • Figure 4: Function values $Q_{\bm{y}}(\bm{x}_{k})$ in \ref{['eg:comparison-1']}.

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 1
  • Theorem 5.1
  • proof
  • Example 2
  • ...and 3 more