{"ID":5443865,"CreatedAt":"2026-07-01T02:07:11.383974684Z","UpdatedAt":"2026-07-03T16:35:57.158869329Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.31951","arxiv_id":"2606.31951","title":"Explicit Series and a Certified Hybrid Evaluator for the $\\ell_p$ Proximity Operator for $0\u003cp\u003c1$","abstract":"The nonconvex $\\ell_p$ quasi-norm with $0\u003cp\u003c1$ is a powerful sparsity surrogate but makes the proximity operator $\\mathrm{prox}_{λ|\\cdot|^p}$ nontrivial to evaluate robustly. We give an explicit characterization of the scalar proximal map for all $0\u003cp\u003c1$, including the threshold structure and conditions ensuring strict, isolated solutions. Applying the Lagrange--Bürmann inversion to the stationarity equation yields a uniformly convergent series for the larger positive root, which provides an exact and numerically stable formula above the classical threshold. We further derive a Mellin--Barnes (MB) integral representation, explaining its radius of convergence and enabling certified truncation. Building on these ingredients, we design a {certified hybrid evaluator} (short series $+$ truncated vertical MB segment) with a computable a priori error bound that remains accurate in the near-threshold regime. For rational $p$, Gauss' multiplication formula reduces the coefficients to finite products of shifted Gamma functions, reorganizing the series into a finite sum of generalized hypergeometric functions and explaining the closed forms at $p\\in\\{1/3,1/2,2/3\\}$. We integrate the evaluator into a proximal-gradient method with an inexact proximal oracle and prove convergence under standard summability of the certificates; MATLAB implementations and numerics confirm accuracy, including near-threshold behavior.","short_abstract":"The nonconvex $\\ell_p$ quasi-norm with $0\u003cp\u003c1$ is a powerful sparsity surrogate but makes the proximity operator $\\mathrm{prox}_{λ|\\cdot|^p}$ nontrivial to evaluate robustly. We give an explicit characterization of the scalar proximal map for all $0\u003cp\u003c1$, including the threshold structure and conditions ensuring strict...","url_abs":"https://arxiv.org/abs/2606.31951","url_pdf":"https://arxiv.org/pdf/2606.31951v1","authors":"[\"Lixin Shen\",\"Jiangyu Yu\"]","published":"2026-06-30T16:56:17Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}