# The Alexander-Hirschowitz theorem for neurovarieties

arXiv: https://arxiv.org/abs/2511.19703 · HTML: https://mlsanity.com/paper/2511.19703/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 23 references that are in the map.
- First posted on arXiv 24 Nov 2025.

## Builds on (most cited first, 16 of 16)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)
- [On the Symmetries of Deep Learning Models and their Internal Representations](https://mlsanity.com/paper/2205.14258/) (2022)
- [Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEs](https://mlsanity.com/paper/2101.08366/) (2021)
- [Pure and Spurious Critical Points: a Geometric Study of Linear Networks](https://mlsanity.com/paper/1910.01671/) (2019)
- [Parameter identifiability of a deep feedforward ReLU neural network](https://mlsanity.com/paper/2112.12982/) (2021)
- [Polynomial Neural Fields for Subband Decomposition and Manipulation](https://mlsanity.com/paper/2302.04862/) (2023)
- [From Identifiable Causal Representations to Controllable Counterfactual Generation: A Survey on Causal Generative Modeling](https://mlsanity.com/paper/2310.11011/) (2023)
- [Augmenting Deep Classifiers with Polynomial Neural Networks](https://mlsanity.com/paper/2104.07916/) (2021)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Activation degree thresholds and expressiveness of polynomial neural networks](https://mlsanity.com/paper/2408.04569/) (2024)
- [On the Geometry and Optimization of Polynomial Convolutional Networks](https://mlsanity.com/paper/2410.00722/) (2024)
- [PolyNet: Polynomial Neural Network for 3D Shape Recognition with PolyShape Representation](https://mlsanity.com/paper/2110.07882/) (2021)
- [Learning on a Razor's Edge: Identifiability and Singularity of Polynomial Neural Networks](https://mlsanity.com/paper/2505.11846/) (2025)
- [The Landscape of Multi-Layer Linear Neural Network From the Perspective of Algebraic Geometry](https://mlsanity.com/paper/2102.04338/) (2021)

## Built on by (most cited first, 6 of 6)

- [Identifiable Equivariant Networks are Layerwise Equivariant](https://mlsanity.com/paper/2601.21645/) (2026)
- [Rational Neural Networks have Expressivity Advantages](https://mlsanity.com/paper/2602.12390/) (2026)
- [The Geometry of Polynomial Group Convolutional Neural Networks](https://mlsanity.com/paper/2603.29566/) (2026)
- [Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects](https://mlsanity.com/paper/2605.09609/) (2026)
- [Singular Learning and Occam's Razor in Deep Monomial Networks](https://mlsanity.com/paper/2606.28464/) (2026)
- [Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks](https://mlsanity.com/paper/2608.27113/) (2026)

## Abstract (arXiv)

We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds di≥ 2ni-1 on the activation degrees imply non defectiveness for any number of outputs, a dimension statement previously obtained from finite identifiability. The proof is based on a direct analysis of the differential of the parameterization. We also investigate secant and Grassmann-secant obstructions outside this range and prove global identifiability for multi-output architectures under the same degree bounds.