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Quasi-Equivalence between Width and Depth of Neural Networks

Fenglei Fan, Rongjie Lai, Ge Wang; 24(183):1−22, 2023.


While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of deep networks. Based on a symmetric consideration, we investigate if the design of artificial neural networks should have a directional preference, and what the mechanism of interaction is between the width and depth of a network. Inspired by the De Morgan law, we address this fundamental question by establishing a quasi-equivalence between the width and depth of ReLU networks. We formulate two transforms for mapping an arbitrary ReLU network to a wide ReLU network and a deep ReLU network respectively, so that the essentially same capability of the original network can be implemented. Based on our findings, a deep network has a wide equivalent, and vice versa, subject to an arbitrarily small error.

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