Deep neural networks (DNNs) are powerful tools for approximating the
distribution of complex data. It is known that data passing through a trained
DNN classifier undergoes a series of geometric and topological simplifications.
While some progress has been made toward understanding these transformations in
neural networks with smooth activation functions, an understanding in the more
general setting of non-smooth activation functions, such as the rectified
linear unit (ReLU), which tend to perform better, is required. Here we propose
that the geometric transformations performed by DNNs during classification
tasks have parallels to those expected under Hamilton's Ricci flow - a tool
from differential geometry that evolves a manifold by smoothing its curvature,
in order to identify its topology. To illustrate this idea, we present a
computational framework to quantify the geometric changes that occur as data
passes through successive layers of a DNN, and use this framework to motivate a
notion of `global Ricci network flow' that can be used to assess a DNN's
ability to disentangle complex data geometries to solve classification
problems. By training more than $1,500$ DNN classifiers of different widths and
depths on synthetic and real-world data, we show that the strength of global
Ricci network flow-like behaviour correlates with accuracy for well-trained
DNNs, independently of depth, width and data set. Our findings motivate the use
of tools from differential and discrete geometry to the problem of
explainability in deep learning.

深度神经网络经过一系列几何和拓扑简化来进行分类任务，而这与 Hamilton 的 Ricci 流在微分几何中平滑曲率以识别拓扑结构的过程存在对应关系。通过构建一个计算框架来量化数据通过 DNN 的不同层时发生的几何变化，我们展示了全局 Ricci 网络流与 DNN 的准确性相关，并且对深度和宽度无关，也适用于不同的数据集。这些发现推动了将微分几何和离散几何工具应用于深度学习解释性问题。