Neural Collapse refers to the curious phenomenon in the end of training of a
neural network, where feature vectors and classification weights converge to a
very simple geometrical arrangement (a simplex). While it has been observed
empirically in various cases and has been theoretically motivated, its
connection with crucial properties of neural networks, like their
generalization and robustness, remains unclear. In this work, we study the
stability properties of these simplices. We find that the simplex structure
disappears under small adversarial attacks, and that perturbed examples "leap"
between simplex vertices. We further analyze the geometry of networks that are
optimized to be robust against adversarial perturbations of the input, and find
that Neural Collapse is a pervasive phenomenon in these cases as well, with
clean and perturbed representations forming aligned simplices, and giving rise
to a robust simple nearest-neighbor classifier. By studying the propagation of
the amount of collapse inside the network, we identify novel properties of both
robust and non-robust machine learning models, and show that earlier, unlike
later layers maintain reliable simplices on perturbed data.

神经坍缩是神经网络训练结束时的奇特现象，它的特征向量和分类权重收敛为一个非常简单的几何结构（简单形）。本研究探究这些简单形的稳定性质，发现它们容易受到小型对抗性攻击的影响，扰动的示例可以在简单形的顶点之间跳跃。此外，作者分析了对抗性扰动优化的网络几何结构，发现神经坍缩在这些情况下普遍存在，干净和扰动的表示形成一致的简单形，并构建了一个鲁棒的近邻分类器。通过研究神经网络内部坍缩量的传播，作者发现了鲁棒和非鲁棒机器学习模型的新特性，并表明早期层次与后期层次在扰动数据上保持可靠的简单形。

神经塌陷的稳健性和稳健性的神经塌陷

On the Robustness of Neural Collapse and the Neural Collapse of  Robustness

A ReLU network is a piecewise linear function over polytopes. Figuring out
the properties of such polytopes is of fundamental importance for the research
and development of neural networks. So far, either theoretical or empirical
studies on polytopes only stay at the level of counting their number, which is
far from a complete characterization of polytopes. To upgrade the
characterization to a new level, here we propose to study the shapes of
polytopes via the number of simplices obtained by triangulating the polytope.
Then, by computing and analyzing the histogram of simplices across polytopes,
we find that a ReLU network has relatively simple polytopes under both
initialization and gradient descent, although these polytopes theoretically can
be rather diverse and complicated. This finding can be appreciated as a novel
implicit bias. Next, we use nontrivial combinatorial derivation to
theoretically explain why adding depth does not create a more complicated
polytope by bounding the average number of faces of polytopes with a function
of the dimensionality. Our results concretely reveal what kind of simple
functions a network learns and its space partition property. Also, by
characterizing the shape of polytopes, the number of simplices be a leverage
for other problems, \textit{e.g.}, serving as a generic functional complexity
measure to explain the power of popular shortcut networks such as ResNet and
analyzing the impact of different regularization strategies on a network's
space partition.

本文利用三角化的方法研究了 ReLU 网络在初始化和梯度下降时的多面体形状，并发现它们相对简单，这是一种新的隐式偏差。此外，本研究还通过界定多面体面的平均数来理论上解释了为什么增加深度不会创建更复杂的多面体，并揭示了网络的简单函数模型和空间分割特性，这些结果具有重要的功能复杂性度量、正则化策略影响等方面的应用潜力。

Deep ReLU 网络拥有令人惊讶的简单多面体

Deep ReLU Networks Have Surprisingly Simple Polytopes

This paper presents $\mathrm{E}(n)$ Equivariant Message Passing Simplicial
Networks (EMPSNs), a novel approach to learning on geometric graphs and point
clouds that is equivariant to rotations, translations, and reflections. EMPSNs
can learn high-dimensional simplex features in graphs (e.g. triangles), and use
the increase of geometric information of higher-dimensional simplices in an
$\mathrm{E}(n)$ equivariant fashion. EMPSNs simultaneously generalize
$\mathrm{E}(n)$ Equivariant Graph Neural Networks to a topologically more
elaborate counterpart and provide an approach for including geometric
information in Message Passing Simplicial Networks. The results indicate that
EMPSNs can leverage the benefits of both approaches, leading to a general
increase in performance when compared to either method. Furthermore, the
results suggest that incorporating geometric information serves as an effective
measure against over-smoothing in message passing networks, especially when
operating on high-dimensional simplicial structures. Last, we show that EMPSNs
are on par with state-of-the-art approaches for learning on geometric graphs.

本文提出了 E (n) 等变消息传递单纯形网络（EMPSNs）的新方法，可以在几何图形和点云上学习，具有旋转、平移和反射等等变性。EMPSNs 可以通过等变的方式使用高维单纯形的几何信息学习高维单纯形特征，是 E (n) 等变图神经网络的拓扑更复杂的扩展，并提供了在消息传递单纯形网络中包含几何信息的方法。结果表明，EMPSNs 可以利用两种方法的优势，相对于任一方法都有更好的性能提升。此外，结果表明，在操作高维单纯形结构时，加入几何信息可以有效防止过度平滑，适用于消息传递网络。最后，我们显示 EMPSNs 与最先进的学习几何图形的方法相当。