Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks

• URL: http://arxiv.org/abs/2505.11846v1
• Date: Sat, 17 May 2025 05:11:17 GMT
• Title: Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks
• Abstract summary: We consider fully-connected networks with activation functions, and focus on the function space they pararize.<n>First, we compute the dimension of the subspace of the neuromanifold parametrized by neuroworks.<n>Second, we show that this subspace is singular.<n>Third, we argue that such singularities often correspond to critical points of the training dynamics.
• Score: 2.9816332334719773
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep neural networks often infer sparse representations, converging to a subnetwork during the learning process. In this work, we theoretically analyze subnetworks and their bias through the lens of algebraic geometry. We consider fully-connected networks with polynomial activation functions, and focus on the geometry of the function space they parametrize, often referred to as neuromanifold. First, we compute the dimension of the subspace of the neuromanifold parametrized by subnetworks. Second, we show that this subspace is singular. Third, we argue that such singularities often correspond to critical points of the training dynamics. Lastly, we discuss convolutional networks, for which subnetworks and singularities are similarly related, but the bias does not arise.

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