Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects

We provide a counterexample to the minimal unimodal conjecture for polynomial neural networks (PNNs) with power activation functions. Fixing the input and output widths, the conjecture states that any minimal filling architecture has unimodal widths for the hidden layers. We found a counterexample via a frontier search and certified it using recursive dimension bounds and symbolic computation. Notably, several subarchitectures of this example exhibit large defect, in contrast with the predominantly small-defect behavior observed in prior examples.

Activation thresholds and expressiveness of polynomial neural networks

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.