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

TL;DR

This paper presents a counterexample to a conjecture about minimal filling architectures in polynomial neural networks, highlighting the existence of architectures with large defect contrary to previous observations.

Contribution

It provides the first counterexample to the minimal unimodal conjecture for polynomial neural networks, using frontier search and symbolic computation.

Findings

Counterexample disproves the minimal unimodal conjecture

Subarchitectures show large defect, contrasting prior small-defect cases

Certified the counterexample with recursive dimension bounds

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.