From Taylor Series to Fourier Synthesis: The Periodic Linear Unit
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916917245116416 |
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| author | Kudo, Shiko |
| author_facet | Kudo, Shiko |
| contents | The dominant paradigm in modern neural networks relies on simple, monotonically-increasing activation functions like ReLU. While effective, this paradigm necessitates large, massively-parameterized models to approximate complex functions. In this paper, we introduce the Periodic Linear Unit (PLU), a learnable sine-wave based activation with periodic non-monotonicity. PLU is designed for maximum expressive power and numerical stability, achieved through its formulation and a paired innovation we term Repulsive Reparameterization, which prevents the activation from collapsing into a non-expressive linear function. We demonstrate that a minimal MLP with only two PLU neurons can solve the spiral classification task, a feat impossible for equivalent networks using standard activations. This suggests a paradigm shift from networks as piecewise Taylor-like approximators to powerful Fourier-like function synthesizers, achieving exponential gains in parameter efficiency by placing intelligence in the neuron itself. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Taylor Series to Fourier Synthesis: The Periodic Linear Unit Kudo, Shiko Machine Learning Numerical Analysis Neural and Evolutionary Computing 68T07 (Primary) 42A10, 41A30, 65D15 (Secondary) I.5.1; G.1.2; G.1.6; I.2.6 The dominant paradigm in modern neural networks relies on simple, monotonically-increasing activation functions like ReLU. While effective, this paradigm necessitates large, massively-parameterized models to approximate complex functions. In this paper, we introduce the Periodic Linear Unit (PLU), a learnable sine-wave based activation with periodic non-monotonicity. PLU is designed for maximum expressive power and numerical stability, achieved through its formulation and a paired innovation we term Repulsive Reparameterization, which prevents the activation from collapsing into a non-expressive linear function. We demonstrate that a minimal MLP with only two PLU neurons can solve the spiral classification task, a feat impossible for equivalent networks using standard activations. This suggests a paradigm shift from networks as piecewise Taylor-like approximators to powerful Fourier-like function synthesizers, achieving exponential gains in parameter efficiency by placing intelligence in the neuron itself. |
| title | From Taylor Series to Fourier Synthesis: The Periodic Linear Unit |
| topic | Machine Learning Numerical Analysis Neural and Evolutionary Computing 68T07 (Primary) 42A10, 41A30, 65D15 (Secondary) I.5.1; G.1.2; G.1.6; I.2.6 |
| url | https://arxiv.org/abs/2508.01175 |