  This paper investigates the relationship between the universal approximation
property of deep neural networks and topological characteristics of datasets.
Our primary contribution is to introduce data topology-dependent upper bounds
on the network width. Specifically, we first show that a three-layer neural
network, applying a ReLU activation function and max pooling, can be designed
to approximate an indicator function over a compact set, one that is
encompassed by a tight convex polytope. This is then extended to a simplicial
complex, deriving width upper bounds based on its topological structure.
Further, we calculate upper bounds in relation to the Betti numbers of select
topological spaces. Finally, we prove the universal approximation property of
three-layer ReLU networks using our topological approach. We also verify that
gradient descent converges to the network structure proposed in our study.
