We analyze energy decay for deep convolutional neural networks employed as
feature extractors, such as Mallat's wavelet scattering transform. For
time-frequency scattering transforms based on Gabor filters, it has been
established that energy decay is exponential, for arbitrary square-integrable
input signals. Our main results allow to prove that this is wrong for wavelet
scattering in arbitrary dimensions. In this setting, the energy decay of the
scattering transform acting on a generic square-integrable signal turns out to
be arbitrarily slow. The fact that this behavior holds for dense subsets of
$L^2(\mathbb{R}^d)$ emphasizes that fast energy decay is generally not a stable
property of signals.
We complement these findings with positive results allowing to conclude fast
(up to exponential) energy decay for generalized Sobolev spaces that are
tailored to the frequency localization of the underlying filter bank.
Both negative and positive results highlight that energy decay in scattering
networks critically depends on the interplay of the respective frequency
localizations of the signal on the one hand, and of the employed filters on the
other.

我们分析应用于特征提取的深度卷积神经网络，如 Mallat 的小波散射变换中的能量衰减。我们的主要结果允许证明，在任意维度的小波散射中，泛型平方可积信号的散射变换的能量衰减可以是任意缓慢的。同时，我们得出积极的结果，允许得出与底层滤波器的频率局部化相适应的广义 Sobolev 空间的快速（指数级）能量衰减。负面和积极的结果都强调了散射网络中的能量衰减严重依赖于信号的频率局部化和所使用的滤波器之间的相互作用。

散射卷积网络中的能量传播可以任意缓慢

Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily  Slow

Extracting and using class-discriminative features is critical for
fine-grained recognition. Existing works have demonstrated the possibility of
applying deep CNNs to exploit features that distinguish similar classes.
However, CNNs suffer from problems including frequency bias and loss of
detailed local information, which restricts the performance of recognizing
fine-grained categories. To address the challenge, we propose a novel texture
branch as complimentary to the CNN branch for feature extraction. We
innovatively utilize Gabor filters as a powerful extractor to exploit texture
features, motivated by the capability of Gabor filters in effectively capturing
multi-frequency features and detailed local information. We implement several
designs to enhance the effectiveness of Gabor filters, including imposing
constraints on parameter values and developing a learning method to determine
the optimal parameters. Moreover, we introduce a statistical feature extractor
to utilize informative statistical information from the signals captured by
Gabor filters, and a gate selection mechanism to enable efficient computation
by only considering qualified regions as input for texture extraction. Through
the integration of features from the Gabor-filter-based texture branch and
CNN-based semantic branch, we achieve comprehensive information extraction. We
demonstrate the efficacy of our method on multiple datasets, including
CUB-200-2011, NA-bird, Stanford Dogs, and GTOS-mobile. State-of-the-art
performance is achieved using our approach.

通过为特征提取引入 Gabor 滤波器作为有力工具来利用纹理特征，并将其与卷积神经网络（CNN）的语义分支相结合，实现全面的信息提取，从而在多个数据集上取得了最先进的性能。

学习 Gabor 纹理特征进行细粒度识别

Learning Gabor Texture Features for Fine-Grained Recognition

In this paper, we aim to improve the mathematical interpretability of
convolutional neural networks for image classification. When trained on natural
image datasets, such networks tend to learn parameters in the first layer that
closely resemble oriented Gabor filters. By leveraging the properties of
discrete Gabor-like convolutions, we prove that, under specific conditions,
feature maps computed by the subsequent max pooling operator tend to
approximate the modulus of complex Gabor-like coefficients, and as such, are
stable with respect to certain input shifts. We then compute a probabilistic
measure of shift invariance for these layers. More precisely, we show that some
filters, depending on their frequency and orientation, are more likely than
others to produce stable image representations. We experimentally validate our
theory by considering a deterministic feature extractor based on the dual-tree
wavelet packet transform, a particular case of discrete Gabor-like
decomposition. We demonstrate a strong correlation between shift invariance on
the one hand and similarity with complex modulus on the other hand.

本文旨在提高卷积神经网络对于图像分类的数学解释性，通过利用离散 Gabor - 类卷积的属性证明特征图在满足某些特定条件下可以逼近复杂 Gabor - 类系数的模，从而具有一定的稳定性，本文通过实验验证了这一理论并与双树小波分组变换一类的离散 Gabor - 类分解进行了比较。