Scale redundancy and soft gauge fixing in positively homogeneous neural networks

Rodrigo Carmo Terin

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arXiv · 2602.14729

Scale redundancy and soft gauge fixing in positively homogeneous neural networks

Rodrigo Carmo Terin

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Abstract

Neural networks with positively homogeneous activations exhibit an exact continuous reparametrization symmetry: neuron-wise rescalings generate parameter-space orbits along which the input--output function is invariant. We interpret this symmetry as a gauge redundancy and introduce gauge-adapted coordinates that separate invariant and scale-imbalance directions. Inspired by gauge fixing in field theory, we introduce a soft orbit-selection (norm-balancing) functional acting only on redundant scale coordinates. We show analytically that it induces dissipative relaxation of imbalance modes to preserve the realized function. In controlled experiments, this orbit-selection penalty expands the stable learning-rate regime and suppresses scale drift without changing expressivity. These results establish a structural link between gauge-orbit geometry and optimization conditioning, providing a concrete connection between gauge-theoretic concepts and machine learning.

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Rodrigo Carmo Terin. 2026-02-16. Scale redundancy and soft gauge fixing in positively homogeneous neural networks. https://arxiv.org/abs/2602.14729

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