Approximation to Smooth Functions by Low-Rank Swish Networks

Zimeng Li, Hongjun Li, Jingyuan Wang, Ke Tang
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:35259-35291, 2025.

Abstract

While deep learning has witnessed remarkable achievements in a wide range of applications, its substantial computational cost imposes limitations on application scenarios of neural networks. To alleviate this problem, low-rank compression is proposed as a class of efficient and hardware-friendly network compression methods, which reduce computation by replacing large matrices in neural networks with products of two small ones. In this paper, we implement low-rank networks by inserting a sufficiently narrow linear layer without bias between each of two adjacent nonlinear layers. We prove that low-rank Swish networks with a fixed depth are capable of approximating any function from the Hölder ball $\mathcal{C}^{\beta, R}([0,1]^d)$ within an arbitrarily small error where $\beta$ is the smooth parameter and $R$ is the radius. Our proposed constructive approximation ensures that the width of linear hidden layers required for approximation is no more than one-third of the width of nonlinear layers, which implies that the computational cost can be decreased by at least one-third compared with a network with the same depth and width of nonlinear layers but without narrow linear hidden layers. Our theoretical finding can offer a theoretical basis for low-rank compression from the perspective of universal approximation theory.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-li25bb, title = {Approximation to Smooth Functions by Low-Rank Swish Networks}, author = {Li, Zimeng and Li, Hongjun and Wang, Jingyuan and Tang, Ke}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {35259--35291}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/li25bb/li25bb.pdf}, url = {https://proceedings.mlr.press/v267/li25bb.html}, abstract = {While deep learning has witnessed remarkable achievements in a wide range of applications, its substantial computational cost imposes limitations on application scenarios of neural networks. To alleviate this problem, low-rank compression is proposed as a class of efficient and hardware-friendly network compression methods, which reduce computation by replacing large matrices in neural networks with products of two small ones. In this paper, we implement low-rank networks by inserting a sufficiently narrow linear layer without bias between each of two adjacent nonlinear layers. We prove that low-rank Swish networks with a fixed depth are capable of approximating any function from the Hölder ball $\mathcal{C}^{\beta, R}([0,1]^d)$ within an arbitrarily small error where $\beta$ is the smooth parameter and $R$ is the radius. Our proposed constructive approximation ensures that the width of linear hidden layers required for approximation is no more than one-third of the width of nonlinear layers, which implies that the computational cost can be decreased by at least one-third compared with a network with the same depth and width of nonlinear layers but without narrow linear hidden layers. Our theoretical finding can offer a theoretical basis for low-rank compression from the perspective of universal approximation theory.} }
Endnote
%0 Conference Paper %T Approximation to Smooth Functions by Low-Rank Swish Networks %A Zimeng Li %A Hongjun Li %A Jingyuan Wang %A Ke Tang %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-li25bb %I PMLR %P 35259--35291 %U https://proceedings.mlr.press/v267/li25bb.html %V 267 %X While deep learning has witnessed remarkable achievements in a wide range of applications, its substantial computational cost imposes limitations on application scenarios of neural networks. To alleviate this problem, low-rank compression is proposed as a class of efficient and hardware-friendly network compression methods, which reduce computation by replacing large matrices in neural networks with products of two small ones. In this paper, we implement low-rank networks by inserting a sufficiently narrow linear layer without bias between each of two adjacent nonlinear layers. We prove that low-rank Swish networks with a fixed depth are capable of approximating any function from the Hölder ball $\mathcal{C}^{\beta, R}([0,1]^d)$ within an arbitrarily small error where $\beta$ is the smooth parameter and $R$ is the radius. Our proposed constructive approximation ensures that the width of linear hidden layers required for approximation is no more than one-third of the width of nonlinear layers, which implies that the computational cost can be decreased by at least one-third compared with a network with the same depth and width of nonlinear layers but without narrow linear hidden layers. Our theoretical finding can offer a theoretical basis for low-rank compression from the perspective of universal approximation theory.
APA
Li, Z., Li, H., Wang, J. & Tang, K.. (2025). Approximation to Smooth Functions by Low-Rank Swish Networks. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:35259-35291 Available from https://proceedings.mlr.press/v267/li25bb.html.

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