Exact Series and Error-Controlled Evaluation of the Nonconvex ℓp Proximity Operator
Abstract
We study the scalar proximity operator of the nonconvex power penalty with exponent between zero and one. Lagrange--Bürmann inversion gives an exact series for the larger stationary root, and the series converges uniformly throughout the active domain of the proximity operator. For rational exponents, an explicit residue-class decomposition reorganizes the coefficients into finite sums of generalized hypergeometric functions, recovering the familiar special cases and producing an additional formula for p=1/3. A Mellin--Barnes formula yields an equivalent residue-corrected representation and analytic truncation bounds. We distinguish the proximal activation threshold from the lower saddle-node threshold and prove that the active proximal branch is uniformly well-conditioned; the practical difficulty near activation is slow series convergence when the exponent approaches one, rather than singular root sensitivity. We also formulate an inexact proximal-gradient step using a computable optimality residual and a descent test. With summable descent errors and vanishing residuals, the objective values converge, the steps vanish, and every cluster point is critical. Tests over a broad parameter grid and in sparse-recovery proximal-gradient runs illustrate both evaluator accuracy and the implementability of the acceptance certificates.
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BibTeX
@misc{shen2026exact,
title = {{Exact Series and Error-Controlled Evaluation of the Nonconvex $\ell_p$ Proximity Operator}},
author = {Lixin Shen and Jiangyu Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.31951},
eprint = {2606.31951},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.31951},
}
RIS
TY - UNPB
TI - Exact Series and Error-Controlled Evaluation of the Nonconvex ℓ_p Proximity Operator
AU - Lixin Shen
AU - Jiangyu Yu
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2606.31951
UR - https://arxiv.org/abs/2606.31951
AB - We study the scalar proximity operator of the nonconvex power penalty with exponent between zero and one. Lagrange--Bürmann inversion gives an exact series for the larger stationary root, and the series converges uniformly throughout the active domain of the proximity operator. For rational exponents, an explicit residue-class decomposition reorganizes the coefficients into finite sums of generalized hypergeometric functions, recovering the familiar special cases and producing an additional formula for p=1/3. A Mellin--Barnes formula yields an equivalent residue-corrected representation and analytic truncation bounds. We distinguish the proximal activation threshold from the lower saddle-node threshold and prove that the active proximal branch is uniformly well-conditioned; the practical difficulty near activation is slow series convergence when the exponent approaches one, rather than singular root sensitivity. We also formulate an inexact proximal-gradient step using a computable optimality residual and a descent test. With summable descent errors and vanishing residuals, the objective values converge, the steps vanish, and every cluster point is critical. Tests over a broad parameter grid and in sparse-recovery proximal-gradient runs illustrate both evaluator accuracy and the implementability of the acceptance certificates.
ER -