  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.
