This paper investigates the computation of proximity operators for scale and signed permutation invariant functions. A scale invariant function remains unchanged under uniform scaling, while a signed permutation invariant function retains its structure despite permutations and sign changes applied to its input variables. Noteworthy examples include the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> function, the ratio of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\ell _1/\ell _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, and its square, with their proximity operators being particularly crucial in sparse signal recovery. We delve into the properties of scale and signed permutation invariant functions, delineating the computation of their proximity operators into three sequential steps: the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\varvec{w}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">w</mi> </mrow> </math></EquationSource> </InlineEquation>-step, r-step, and d-step. These steps collectively form a procedure termed as WRD, with the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\varvec{w}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">w</mi> </mrow> </math></EquationSource> </InlineEquation>-step being of utmost importance and requiring careful treatment. Leveraging this procedure, we present a method for explicitly and efficiently computing the proximity operator of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\ell _1/\ell _2)^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and introduce an algorithm for the proximity operator of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell _1/\ell _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Numerical experiments on sparse signal recovery corroborate the analysis and show that first-order methods equipped with these proximity operators outperform <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-based baselines in reconstruction accuracy.

Computing Proximity Operators of Scale and Signed Permutation Invariant Functions

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2025年11月4日

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Jianqing Jia,
Ashley Prater-Bennette,
Lixin Shen

摘要

This paper investigates the computation of proximity operators for scale and signed permutation invariant functions. A scale invariant function remains unchanged under uniform scaling, while a signed permutation invariant function retains its structure despite permutations and sign changes applied to its input variables. Noteworthy examples include the $\ell _0$ $ℓ_{0}$ function, the ratio of $\ell _1/\ell _2$ $ℓ_{1} / ℓ_{2}$ , and its square, with their proximity operators being particularly crucial in sparse signal recovery. We delve into the properties of scale and signed permutation invariant functions, delineating the computation of their proximity operators into three sequential steps: the ${\varvec{w}}$ $w$ -step, r-step, and d-step. These steps collectively form a procedure termed as WRD, with the ${\varvec{w}}$ $w$ -step being of utmost importance and requiring careful treatment. Leveraging this procedure, we present a method for explicitly and efficiently computing the proximity operator of $(\ell _1/\ell _2)^2$ ${(ℓ_{1} / ℓ_{2})}^{2}$ and introduce an algorithm for the proximity operator of $\ell _1/\ell _2$ $ℓ_{1} / ℓ_{2}$ . Numerical experiments on sparse signal recovery corroborate the analysis and show that first-order methods equipped with these proximity operators outperform $\ell _1$ $ℓ_{1}$ -based baselines in reconstruction accuracy.