Data Topology-Dependent Upper Bounds of Neural Network Widths
This provides theoretical insights into neural network design for researchers in machine learning, though it appears incremental as it builds on existing approximation theory with a topological focus.
The paper tackles the problem of relating neural network universal approximation to dataset topology by introducing data topology-dependent upper bounds on network width, proving that a three-layer ReLU network with max pooling can approximate indicator functions and achieve universal approximation based on topological structures like Betti numbers.
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.