稀疏各向同性正则化用于球面谐波表示的随机场
Sparse Isotropic Regularization for Spherical Harmonic Representations of Random Fields on the Sphere
AI总结：
本文研究了球面上随机场的稀疏各向同性正则化方法，通过混合l1和l2范数的正则化项保持各向同性，结合CMB数据验证了正则化与误差之间的平衡。
摘要：
本文讨论了单位球面S²上随机场的稀疏各向同性正则化问题，其中场以球谐基函数展开。关键特征是正则化项中混合了l1和l2范数的范数，使得正则化保持各向同性，即如果观测到的随机场是强各向同性的，则正则化后的场也如此。帕累托有效前沿用于显示稀疏诱导范数与数据偏差项之间的权衡，以帮助选择合适的正则化参数。详细讨论了使用宇宙微波背景(CMB)数据的数值示例。特别是，数值结果探讨了正则化与偏差之间的权衡，并显示可以实现显著的稀疏性同时保持小的L2误差。
英文摘要：
This paper discusses sparse isotropic 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.
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