From Taylor Series to Fourier Synthesis: The Periodic Linear Unit

Shiko Kudo

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arXiv · 2508.01175

From Taylor Series to Fourier Synthesis: The Periodic Linear Unit

Shiko Kudo

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Abstract

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.

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Shiko Kudo. 2025-08-26. From Taylor Series to Fourier Synthesis: The Periodic Linear Unit. https://arxiv.org/abs/2508.01175

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