Activation thresholds and expressiveness of polynomial neural networks

• URL: http://arxiv.org/abs/2408.04569v1
• Date: Thu, 8 Aug 2024 16:28:56 GMT
• Title: Activation thresholds and expressiveness of polynomial neural networks
• Authors: Bella Finkel, Jose Israel Rodriguez, Chenxi Wu, Thomas Yahl,
• Abstract summary: Polynomial neural networks have been implemented in a range of applications. In this work, we introduce the notion of the activation threshold of a network architecture.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Polynomial neural networks have been implemented in a range of applications and present an advantageous framework for theoretical machine learning. A polynomial neural network of fixed architecture and activation degree gives an algebraic map from the network's weights to a set of polynomials. The image of this map is the space of functions representable by the network. Its Zariski closure is an affine variety known as a neurovariety. The dimension of a polynomial neural network's neurovariety provides a measure of its expressivity. In this work, we introduce the notion of the activation threshold of a network architecture which expresses when the dimension of a neurovariety achieves its theoretical maximum. In addition, we prove expressiveness results for polynomial neural networks with equi-width~architectures.

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