A Continuous Transform for Localized Ridgelets

Joseph Shenouda, Rahul Parhi, Robert D Nowak

Abstract: We develop a new continuous wavelet-like transform for localized ridgelets. Contrary to the classical ridgelets (which are not local), this new dictionary exhibits desirable decay so that all the atoms lie in $L^2(\mathbb{R}^d)$. Furthermore, each localized ridgelet atom is itself a superposition of continuously many classical ridgelets. Our construction hinges on a careful wavelet analysis in the Radon domain, different than the usual Radon-domain wavelet analysis found in the study of classical ridgelets. This is crucial in ensuring the locality of our new, localized ridgelet atoms. We prove a continuous transform and inversion formula for this new dictionary. Finally, due to the locality of these atoms, we conjecture that this new dictionary is better conditioned than the system of non-local ridge functions used ubiquitously in modern neural networks.

Submission Type: Full Paper