We describe the emergence of a Convolution Bottleneck (CBN) structure in
CNNs, where the network uses its first few layers to transform the input
representation into a representation that is supported only along a few
frequencies and channels, before using the last few layers to map back to the
outputs. We define the CBN rank, which describes the number and type of
frequencies that are kept inside the bottleneck, and partially prove that the
parameter norm required to represent a function $f$ scales as depth times the
CBN rank $f$. We also show that the parameter norm depends at next order on the
regularity of $f$. We show that any network with almost optimal parameter norm
will exhibit a CBN structure in both the weights and - under the assumption
that the network is stable under large learning rate - the activations, which
motivates the common practice of down-sampling; and we verify that the CBN
results still hold with down-sampling. Finally we use the CBN structure to
interpret the functions learned by CNNs on a number of tasks.

卷积瓶颈 (CBN) 结构在卷积神经网络 (CNNs) 中的出现，说明网络使用前几层将输入表示转化为仅在少数频率和通道上支持的表示，然后使用最后几层将其映射回输出。我们定义了 CBN 秩，用于描述在瓶颈中保留的频率数量和类型，并在某种程度上证明了表示函数 $f$ 所需的参数范数与深度乘以 CBN 秩 $f$ 成正比。我们还展示了参数范数对 $f$ 的规则性的依赖关系。我们表明，任何具有近乎最优参数范数的网络都会在权重和激活函数中表现出 CBN 结构，这也验证了下采样的常见做法，并验证了 CBN 结果在使用下采样时仍然成立。最后，我们使用 CBN 结构对 CNN 在多个任务中学到的函数进行了解释。

CNN 需要哪些频率？特征学习中的新型瓶颈结构

Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature  Learning

While several convolution-like operators have recently been proposed for
extracting features out of point clouds, down-sampling an unordered point cloud
in a deep neural network has not been rigorously studied. Existing methods
down-sample the points regardless of their importance for the output. As a
result, some important points in the point cloud may be removed, while less
valuable points may be passed to the next layers. In contrast, adaptive
down-sampling methods sample the points by taking into account the importance
of each point, which varies based on the application, task and training data.
In this paper, we propose a permutation-invariant learning-based adaptive
down-sampling layer, called Critical Points Layer (CPL), which reduces the
number of points in an unordered point cloud while retaining the important
points. Unlike most graph-based point cloud down-sampling methods that use
$k$-NN search algorithm to find the neighbouring points, CPL is a global
down-sampling method, rendering it computationally very efficient. The proposed
layer can be used along with any graph-based point cloud convolution layer to
form a convolutional neural network, dubbed CP-Net in this paper. We introduce
a CP-Net for $3$D object classification that achieves the best accuracy for the
ModelNet$40$ dataset among point cloud-based methods, which validates the
effectiveness of the CPL.

该研究提出了一种自适应化下采样方法来保留点云中的重要点，使其可以与任何基于图的点云卷积层结合成为一个卷积神经网络用于 3D 物体分类。该方法在点云类数据集 ModelNet40 中取得了最佳结果。

点云分类的自适应分层下采样

Adaptive Hierarchical Down-Sampling for Point Cloud Classification

We integrate the recently proposed spatial transformer network (SPN)
[Jaderberg et. al 2015] into a recurrent neural network (RNN) to form an
RNN-SPN model. We use the RNN-SPN to classify digits in cluttered MNIST
sequences. The proposed model achieves a single digit error of 1.5% compared to
2.9% for a convolutional networks and 2.0% for convolutional networks with SPN
layers. The SPN outputs a zoomed, rotated and skewed version of the input
image. We investigate different down-sampling factors (ratio of pixel in input
and output) for the SPN and show that the RNN-SPN model is able to down-sample
the input images without deteriorating performance. The down-sampling in
RNN-SPN can be thought of as adaptive down-sampling that minimizes the
information loss in the regions of interest. We attribute the superior
performance of the RNN-SPN to the fact that it can attend to a sequence of
regions of interest.

本文研究了如何将最近提出的空间变换网络（SPN）[Jaderberg et. al 2015] 集成到递归神经网络（RNN）中，形成一个 RNN-SPN 模型，用于对杂乱的 MNIST 序列中的数字进行分类。我们的模型通过对感兴趣的区域进行自适应降采样，从而取得了 1.5% 的错误率，表现优于使用卷积神经网络或卷积神经网络加上 SPN 层的方法。