Stability of the scattering transform for deformations with minimal regularity

IF 2.1 1区 数学 Q1 MATHEMATICS
Fabio Nicola, S. Ivan Trapasso
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引用次数: 0

Abstract

The wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of harmonic and multiscale analysis can be ingeniously exploited to build a signal representation with provable geometric stability properties, such as the Lipschitz continuity to the action of small C2 diffeomorphisms – a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale Cα, α>0. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class Cα, α>1, whereas instability phenomena can occur at lower regularity levels modeled by Cα, 0α<1. While the analysis at the threshold given by Lipschitz (or even C1) regularity remains beyond reach, we are able to prove a stability bound in that case, up to ε losses.

最小规则变形散射变换的稳定性
stsamphane Mallat引入的小波散射变换是一个独特的例子,说明谐波和多尺度分析的思想如何被巧妙地利用来构建具有可证明的几何稳定性特性的信号表示,例如小C2微分同态作用的Lipschitz连续性——这在理论和实际目的上都是一个显著的结果,本质上取决于滤波器的选择和它们在层次结构中的排列。在本文中,我们在Hölder规则尺度Cα, α>0中进一步研究了散射结构与变形规则之间的密切关系。我们能够精确地识别稳定阈值,证明对于Cα, α>1类变形仍然可以实现稳定,而Cα, 0≤α<1模型中的不稳定现象可能在更低的规则水平上发生。虽然Lipschitz(甚至C1)正则性给出的阈值的分析仍然遥不可及,但我们能够证明在这种情况下的稳定性界,直到ε损失。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
4.30
自引率
0.00%
发文量
84
审稿时长
6 months
期刊介绍: Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.
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