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 $C^\alpha$, $\alpha >0$. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class $C^{\alpha}$, $\alpha>1$, whereas instability phenomena can occur at lower regularity levels modelled by $C^\alpha$, $0\le \alpha <1$. 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 $\varepsilon$ losses.
翻译:在深革命神经网络的数学分析中,St'ephane Mallat引进的波粒散射变异是一个独特的例子,说明如何将多尺度分析的理念与一系列非线性模量分析相结合,以可变几何稳定性特性建立一个非加速的、翻译变异信号表示,即:利普西茨对小的C$2美元二异己形态行动的连续性,这在理论和实践上都是一个显著的结果,这主要取决于过滤器的选择和它们进入等级结构的安排。在本说明中,我们进一步调查了分散结构与H\"老式常规规模中变形的规律性之间的亲密关系,即Cäalpha$2美元, 美元=alphephephistisms的动作。我们能够准确地确定稳定性门槛,证明对于小的Cäalpha $1美元(美元)的变形仍然可以实现稳定,而不稳定现象则发生在以美元为模型的低常规水平, $0\lesi $1美元, 直至常规值1美元(美元) lequal) listal deal case cust a custive cas be lex be lex lex ex be ex lex lexilate a pill be pill bexild a lexilate axild axxild a lex lex lexxxxxild a lexild a lexilt lexiltiltalitalit.