Stability of the scattering transform for deformations with minimal regularity
Stability of the scattering transform for deformations with minimal regularity
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具有最小规律性的变形的散射变换的稳定性
DOI：
10.48550/arxiv.2205.11142
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
S. I. Trapasso
中科院分区：
文献类型：
--
作者：
F. Nicola;S. I. Trapasso
。在深度卷积神经网络的数学分析中，ST‘ephane Mallat引入的小波散射变换是一个独特的例子，说明了如何将多尺度分析的思想与一系列模非线性项相结合，以构建具有可证明的几何稳定性的非扩张、平移不变的信号表示，即小C2 diff同构的作用的Lipschitz连续性-这是一个显著的理论和实践目的，内在地取决于fi的选择和它们在分层结构中的排列。在本文中，我们进一步研究了H？旧正则标度Cα，α>；0中的散射结构与形变的正则性之间的密切关系。我们能够精确地识别稳定性阈值，证明对于C级α，α&>1的变形仍然可以实现稳定性，而不稳定现象可能发生在由Cα，0≤α；1模拟的较低的规则性水平。虽然在由利普希茨(甚至是C1)规则性给出的阈值下的行为仍然是遥不可及的，但我们能够证明在这种情况下的稳定性界限，直到ε损失。
. Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by St´ephane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small C 2 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 H¨older regularity scale C α , α > 0. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class C α , α > 1, whereas instability phenomena can occur at lower regularity levels modelled by C α , 0 ≤ α < 1. While the behaviour at the threshold given by Lipschitz (or even C 1 ) regularity remains beyond reach, we are able to prove a stability bound in that case, up to ε losses.
影响因子：
2.5
作者：
Dongmian Zou;R. Balan;Maneesh Kumar Singh
通讯作者：
Dongmian Zou;R. Balan;Maneesh Kumar Singh
影响因子：
3
作者：
Song Mei;A. Montanari
通讯作者：
Song Mei;A. Montanari
DOI：
10.1073/pnas.1903070116
发表时间：
2019-08-06
影响因子：
11.1
作者：
Belkin, Mikhail;Hsu, Daniel;Mandal, Soumik
通讯作者：
Mandal, Soumik