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The Convolutional Neural Network (CNN) is one of the most prominent neuralnetwork architectures in deep learning. Despite its widespread adoption, ourunderstanding of its universal approximation properties has been limited due toits intricate nature. CNNs inherently function as tensor-to-tensor mappings,preserving the spatial structure of input data. However, limited research hasexplored the universal approximation properties of fully convolutional neuralnetworks as arbitrary continuous tensor-to-tensor functions. In this study, wedemonstrate that CNNs, when utilizing zero padding, can approximate arbitrarycontinuous functions in cases where both the input and output values exhibitthe same spatial shape. Additionally, we determine the minimum depth of theneural network required for approximation and substantiate its optimality. Wealso verify that deep, narrow CNNs possess the UAP as tensor-to-tensorfunctions. The results encompass a wide range of activation functions, and ourresearch covers CNNs of all dimensions.

Universal Approximation Property of Fully Convolutional Neural Networks with Zero Padding