Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functions

• URL: http://arxiv.org/abs/2301.04605v1
• Date: Wed, 11 Jan 2023 17:57:33 GMT
• Title: Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functions
• Authors: Ido Ben-Shaul, Tomer Galanti and Shai Dekel
• Abstract summary: We consider two classes of target functions: generalized bandlimited functions and Sobolev-Type balls. Our results demonstrate that multiplicative neural networks can approximate these functions with significantly fewer layers and neurons. These findings suggest that multiplicative gates can outperform standard feed-forward layers and have potential for improving neural network design.
• Score: 9.936974568429173
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Multiplication layers are a key component in various influential neural network modules, including self-attention and hypernetwork layers. In this paper, we investigate the approximation capabilities of deep neural networks with intermediate neurons connected by simple multiplication operations. We consider two classes of target functions: generalized bandlimited functions, which are frequently used to model real-world signals with finite bandwidth, and Sobolev-Type balls, which are embedded in the Sobolev Space $\mathcal{W}^{r,2}$. Our results demonstrate that multiplicative neural networks can approximate these functions with significantly fewer layers and neurons compared to standard ReLU neural networks, with respect to both input dimension and approximation error. These findings suggest that multiplicative gates can outperform standard feed-forward layers and have potential for improving neural network design.

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