Universal Approximation Properties for an ODENet and a ResNet: Mathematical Analysis and Numerical Experiments

• URL: http://arxiv.org/abs/2101.10229v3
• Date: Thu, 18 May 2023 01:22:34 GMT
• Title: Universal Approximation Properties for an ODENet and a ResNet: Mathematical Analysis and Numerical Experiments
• Authors: Yuto Aizawa, Masato Kimura, and Kazunori Matsui
• Abstract summary: We prove a universal approximation property (UAP) for a class of ODENet and a class of ResNet. We use this to construct a learning algorithm for ODENet.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove a universal approximation property (UAP) for a class of ODENet and a class of ResNet, which are simplified mathematical models for deep learning systems with skip connections. The UAP can be stated as follows. Let $n$ and $m$ be the dimension of input and output data, and assume $m\leq n$. Then we show that ODENet of width $n+m$ with any non-polynomial continuous activation function can approximate any continuous function on a compact subset on $\mathbb{R}^n$. We also show that ResNet has the same property as the depth tends to infinity. Furthermore, we derive the gradient of a loss function explicitly with respect to a certain tuning variable. We use this to construct a learning algorithm for ODENet. To demonstrate the usefulness of this algorithm, we apply it to a regression problem, a binary classification, and a multinomial classification in MNIST.

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