Multivariate $\alpha$-molecules

TL;DR

This paper extends the $eta$-molecules framework to higher dimensions, unifying various directional systems for multivariate data and demonstrating near-optimal sparse approximation of 3D signals like videos.

Contribution

It generalizes the $eta$-molecules theory from 2D to higher dimensions, enabling unified analysis of multivariate directional systems.

Findings

Cross-Gramian of $eta$-molecules systems is localized in higher dimensions

Derived near-optimal approximation rates for 3D data such as videos

Unified framework encompasses ridgelets, curvelets, shearlets, and more

Abstract

The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by ridgelets, curvelets, and shearlets, to name a few. The great variety of such so-called directional systems motivated the search for a common framework, which unites many under one roof and enables a simultaneous analysis, for example with respect to approximation properties. Building on the concept of parabolic molecules, the recently introduced framework of $\alpha$-molecules does in fact include the previous mentioned systems. Until now however it is confined to the bivariate setting, whereas nowadays one often deals with higher dimensional data. This motivates the extension of this unifying theory to dimensions larger than 2, put forward in this work. In particular, we generalize the central result that the cross-Gramian of any two systems of $\alpha$-molecules will to some extent be localized. As an exemplary application, we investigate the sparse approximation of video signals, which are instances of 3D data. The multivariate theory allows us to derive almost optimal approximation rates for a large class of representation systems.