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

arXiv: https://arxiv.org/abs/2605.09609 · HTML: https://mlsanity.com/paper/2605.09609/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 21 references that are in the map.
- First posted on arXiv 10 May 2026.

## Builds on (most cited first, 13 of 13)

- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)
- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [Geometry of Linear Convolutional Networks](https://mlsanity.com/paper/2108.01538/) (2021)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Geometry of Lightning Self-Attention: Identifiability and Dimension](https://mlsanity.com/paper/2408.17221/) (2024)
- [Activation degree thresholds and expressiveness of polynomial neural networks](https://mlsanity.com/paper/2408.04569/) (2024)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)
- [The Alexander-Hirschowitz theorem for neurovarieties](https://mlsanity.com/paper/2511.19703/) (2025)
- [A Training Framework for Optimal and Stable Training of Polynomial Neural Networks](https://mlsanity.com/paper/2505.11589/) (2025)

## Built on by (most cited first, 1 of 1)

- [Singular Learning and Occam's Razor in Deep Monomial Networks](https://mlsanity.com/paper/2606.28464/) (2026)

## Abstract (arXiv)

We provide counterexamples to the unimodal minimal filling architecture 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 counterexamples via a frontier search, recursive dimension bounds on neurovarieties, and symbolic computation. Notably, several subarchitectures of our main example exhibit large defect, in contrast with the predominantly small-defect behavior observed in prior literature.