From Taylor Series to Fourier Synthesis: The Periodic Linear Unit

• URL: http://arxiv.org/abs/2508.01175v1
• Date: Sat, 02 Aug 2025 03:26:48 GMT
• Title: From Taylor Series to Fourier Synthesis: The Periodic Linear Unit
• Authors: Shiko Kudo,
• Abstract summary: We introduce the Periodic Linear Unit (PLU), a learnable sine-wave based activation with periodic non-monotonicity.<n>We demonstrate that a minimal activation with only two neurons can solve the spiral classification task, a feat impossible for equivalent networks using standard activations.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• 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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