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

What the math gives to ML

The paper supplies a numerically careful scalar proximal map for the nonconvex sparsity penalty |x|^p, 0<p<1, including the discontinuous threshold at which the global minimizer leaves zero. This is transferable to neural-network optimization as an exact or certified inexact proximal step for inducing genuinely sparse weights, rather than using an ordinary differentiable approximation to an l_p penalty. The most useful engineering asset is the combination of a closed-form threshold, a rapidly evaluable Lagrange-inversion series away from the threshold, and a safeguarded root evaluator near the threshold. A practical first target is a proximal-gradient or alternating optimizer for sparse MLP weights, convolution kernels, or MoE router parameters.