In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as ${\mathbf{L}}^q({\mathbb{R}}^n)$ norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of $C^2$ diffeomorphisms in specific cases. Lastly, we extend our results to formulate an operator invariant to the action of rotations $R\in \text{SO}(n)$ and an operator that is equivariant to the action of rotations of $R\in \text{SO}(n)$.

中文翻译：

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关于非加窗散射变换的推广

在本文中，我们推广了有限深度小波散射变换，将其表示为${\mathbf{L}}^q（{\mathbb{右}}^n）$连续小波变换（或二进小波变换）和收缩非线性级联的范数。然后我们为这些算子提供规范，证明这些算子是明确定义的，并且 Lipschitz 连续于$C^2$特定情况下的微分同态。最后，我们扩展我们的结果来制定一个对旋转作用不变的算子$右\epsilon \text{所以}（n）$和一个与旋转动作等价的算子$右\epsilon \text{所以}（n）$。