This paper discusses isotropic sparse regularization for a random field on the unit sphere ${\mathbb{S}}^2$ in ${\mathbb{R}}^3$, where the field is expanded in terms of a spherical harmonic basis. A key feature is that the norm used in the regularization term, a hybrid of the ${\ell }_1$ and ${\ell }_2$-norms, is chosen so that the regularization preserves isotropy, in the sense that if the observed random field is strongly isotropic then so too is the regularized field. The Pareto efficient frontier is used to display the trade-off between the sparsity-inducing norm and the data discrepancy term, in order to help in the choice of a suitable regularization parameter. A numerical example using Cosmic Microwave Background (CMB) data is considered in detail. In particular, the numerical results explore the trade-off between regularization and discrepancy, and show that substantial sparsity can be achieved along with small $L_2$ error.

本文讨论了单位球面上随机场的各向同性稀疏正则化 ${\mathbb{小号}}^2$ 在 ${\mathbb{[R}}^3$，其中场以球形谐波为基础扩展。一个关键特征是正则化术语中使用的规范是${\ell }_{\mathrm{1个}}$ 和 ${\ell }_2$选择-范数以使正则化保持各向同性，在某种意义上，如果观察到的随机场是强各向同性的，则正则化场也是如此。帕累托有效边界用于显示稀疏诱导范数和数据差异项之间的权衡，以帮助选择合适的正则化参数。详细考虑了使用宇宙微波背景（CMB）数据的数值示例。尤其是，数值结果探索了正则化和差异之间的权衡，并表明可以实现相当大的稀疏性，同时获得较小的${\mathrm{大号}}_2$ 错误。