The Stochastic Properties of ${\ell }^1$-Regularized Spherical Gaussian Fields

Abstract

Convex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an important question is thus whether such regularization techniques preserve the properties of the underlying probability measures. We focus here on a case which has produced a very lively debate in the cosmological literature, namely Gaussian and isotropic spherical random fields, and we prove that Gaussianity and isotropy are not conserved in general under convex regularization over a Fourier dictionary, such as the orthonormal system of spherical harmonics.

References in corpus (3)

• Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results
• Low-l CMB Analysis and Inpainting
• Sparse image reconstruction on the sphere: implications of a new sampling theorem

Cited by in corpus (1)

• Needlet thresholding methods in component separation