多元$α$ -分子
Multivariate $α$-molecules
摘要：
小波的次优性能方面的多变量数据的近似产生了新的表示系统，专门设计的数据与各向异性功能。一些突出的例子是脊波、曲波和剪切波，仅举几例。各种各样的这种所谓的定向系统激发了寻找一个共同的框架，它将许多人团结在一个屋檐下，并使同时分析，例如关于近似性质。建立在抛物线分子的概念上，最近引入的$α$ -分子的框架实际上包括前面提到的系统。然而，到目前为止，它仅限于双变量设置，而现在人们经常处理高维数据。这促使扩展这一统一理论的尺寸大于2，在这项工作中提出。特别是，我们推广了中心的结果，任何两个系统的$α$ -分子的交叉Gramian将在一定程度上本地化。作为一个示例性的应用程序，我们调查的稀疏近似的视频信号，这是3D数据的实例。多元理论使我们能够得到几乎最优逼近率的一大类表示系统。
英文摘要：
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 $α$-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 $α$-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.
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