  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.
