On Enhancing Expressive Power via Compositions of Single Fixed-Size ReLU Network

• URL: http://arxiv.org/abs/2301.12353v2
• Date: Tue, 30 May 2023 22:01:05 GMT
• Title: On Enhancing Expressive Power via Compositions of Single Fixed-Size ReLU Network
• Authors: Shijun Zhang, Jianfeng Lu, Hongkai Zhao
• Abstract summary: We show that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent.
• Score: 11.66117393949175
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that $\mathcal{L}_2\circ \boldsymbol{g}^{\circ r}\circ \boldsymbol{\mathcal{L}}_1$ can approximate $1$-Lipschitz continuous functions on $[0,1]^d$ with an error $\mathcal{O}(r^{-1/d})$, where $\boldsymbol{g}$ is realized by a fixed-size ReLU network, $\boldsymbol{\mathcal{L}}_1$ and $\mathcal{L}_2$ are two affine linear maps matching the dimensions, and $\boldsymbol{g}^{\circ r}$ denotes the $r$-times composition of $\boldsymbol{g}$. Furthermore, we extend such a result to generic continuous functions on $[0,1]^d$ with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

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