Explicit Series and a Certified Hybrid Evaluator for the $\ell_p$ Proximity Operator for $0<p<1$

The nonconvex $\ell_p$ quasi-norm with $0<p<1$ 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 $0<p<1$, 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 exa...