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

2026-05-10

AI Summary

This study investigates the conjecture that minimal-width polynomial neural networks (PNNs) must exhibit a unimodal width profile across hidden layers. By integrating advanced search strategies, recursive dimensional bound analysis, and symbolic computation, the authors construct the first counterexample that definitively refutes this unimodality assumption. The discovered architecture features a notably large defective substructure, challenging the prevailing belief that optimal PNNs are dominated by small defects. This finding reveals richer and more complex patterns in the optimal distribution of hidden layer widths, thereby expanding our understanding of the representational capacity of polynomial neural networks.

Abstract

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.

Problem

• polynomial neural networks
• minimal filling architectures
• unimodal conjecture
• defect
• frontier search

Innovation

• polynomial neural networks
• minimal filling architecture
• unimodal conjecture
• frontier search
• defect