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

Data Topology-Dependent Upper Bounds of Neural Network Widths