The Alexander-Hirschowitz theorem for neurovarieties

arXiv:2511.19703 · math.AG, cs.AI, cs.LG, math.AC · Submitted 2025-11-24 · Read on arXiv

math.AG, cs.AI, cs.LG, math.AC

Submitted: 2025-11-24

Updated: 2026-09-01

Comments: 15 pages. Thanks to Kevin Dao, we identified a gap in the previous proof of the main non-defectiveness result. We replaced the transversality argument with a direct analysis of the differential, based on a new linear independence lemma, obtaining the degree bound $d_i\ge 2n_i-1$

Code: https://github.com/msslxa/Neurovarieties

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Polynomial Neural Networks — 12× in this paper
• Activation Function — 7× in this paper · explained in 1 episode
• Embeddings — 4× in this paper · explained in 2 episodes
• Generalization — 2× in this paper · explained in 13 episodes
• Interpretability — 2× in this paper · explained in 15 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Degeneracy — 1× in this paper · explained in 3 episodes
• Generative Modeling — 1× in this paper · explained in 3 episodes
• Parameter Identifiability — 1× in this paper · explained in 2 episodes
• Pruning — 1× in this paper · explained in 4 episodes
• Redundancy — 1× in this paper · explained in 2 episodes
• Regularization — 1× in this paper · explained in 3 episodes
• Representation Learning — 1× in this paper · explained in 2 episodes
• Robustness — 1× in this paper · explained in 5 episodes
• Sample Complexity — 1× in this paper · explained in 15 episodes
• Symbolic Regression — 1× in this paper · explained in 5 episodes

Sources

• Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
• Geometry of Lightning Self-Attention: Identifiability and Dimension
• Identifiability of Deep Polynomial Neural Networks

Related papers

• Algebraic Invariants of Lightning Self-Attention