Neural networks can be thought of as applying a transformation to an input
dataset. The way in which they change the topology of such a dataset often
holds practical significance for many tasks, particularly those demanding
non-homeomorphic mappings for optimal solutions, such as classification
problems. In this work, we leverage the fact that neural networks are
equivalent to continuous piecewise-affine maps, whose rank can be used to
pinpoint regions in the input space that undergo non-homeomorphic
transformations, leading to alterations in the topological structure of the
input dataset. Our approach enables us to make use of the relative homology
sequence, with which one can study the homology groups of the quotient of a
manifold $\mathcal{M}$ and a subset $A$, assuming some minimal properties on
these spaces.
As a proof of principle, we empirically investigate the presence of low-rank
(topology-changing) affine maps as a function of network width and mean weight.
We show that in randomly initialized narrow networks, there will be regions in
which the (co)homology groups of a data manifold can change. As the width
increases, the homology groups of the input manifold become more likely to be
preserved. We end this part of our work by constructing highly non-random wide
networks that do not have this property and relating this non-random regime to
Dale's principle, which is a defining characteristic of biological neural
networks.
Finally, we study simple feedforward networks trained on MNIST, as well as on
toy classification and regression tasks, and show that networks manipulate the
topology of data differently depending on the continuity of the task they are
trained on.

神经网络的拓扑结构及同调群，以及仿射映射与任务连续性的关系的研究。

神经表征的拓扑分类的秩分解

A rank decomposition for the topological classification of neural  representations

In a seminal paper, Erdos and Renyi identified the threshold for connectivity
of the random graph G(n,p). In particular, they showed that if p >> log(n)/n
then G(n,p) is almost always connected, and if p << log(n)/n then G(n,p) is
almost always disconnected, as n goes to infinity.
The clique complex X(H) of a graph H is the simplicial complex with all
complete subgraphs of H as its faces. In contrast to the zeroth homology group
of X(H), which measures the number of connected components of H, the higher
dimensional homology groups of X(H) do not correspond to monotone graph
properties. There are nevertheless higher dimensional analogues of the
Erdos-Renyi Theorem.
We study here the higher homology groups of X(G(n,p)). For k > 0 we show the
following. If p = n^alpha, with alpha < -1/k or alpha > - 1/(2k+1), then the
kth homology group of X(G(n,p)) is almost always vanishing, and if -1/k < alpha
< -1/(k+1), then it is almost always nonvanishing.
We also give estimates for the expected rank of homology, and exhibit
explicit nontrivial classes in the nonvanishing regime. These estimates suggest
that almost all d-dimensional clique complexes have only one nonvanishing
dimension of homology, and we cannot rule out the possibility that they are
homotopy equivalent to wedges of spheres.

研究了随机图的连通性门槛以及其高阶同调群，得出了关于其维度的性质，同时对于零维同调群的严格计算提供了定量估计，并在非零维同调群中展现了明显的非平凡特征。