# Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks

arXiv: https://arxiv.org/abs/2608.27113 · HTML: https://mlsanity.com/paper/2608.27113/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Hidden symmetries of ReLU networks](https://mlsanity.com/paper/2306.06179/) (2023)
- [Pure and Spurious Critical Points: a Geometric Study of Linear Networks](https://mlsanity.com/paper/1910.01671/) (2019)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Affine symmetries and neural network identifiability](https://mlsanity.com/paper/2006.11727/) (2020)
- [Geometry of Lightning Self-Attention: Identifiability and Dimension](https://mlsanity.com/paper/2408.17221/) (2024)
- [Neural network identifiability for a family of sigmoidal nonlinearities](https://mlsanity.com/paper/1906.06994/) (2019)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [On the Geometry and Optimization of Polynomial Convolutional Networks](https://mlsanity.com/paper/2410.00722/) (2024)
- [Learning on a Razor's Edge: Identifiability and Singularity of Polynomial Neural Networks](https://mlsanity.com/paper/2505.11846/) (2025)
- [Identifiable Equivariant Networks are Layerwise Equivariant](https://mlsanity.com/paper/2601.21645/) (2026)
- [The Alexander-Hirschowitz theorem for neurovarieties](https://mlsanity.com/paper/2511.19703/) (2025)
- [Most ReLU Networks Admit Identifiable Parameters](https://mlsanity.com/paper/2605.03601/) (2026)
- [The Symmetries of Three-Layer ReLU Networks](https://mlsanity.com/paper/2605.18319/) (2026)

## Abstract (arXiv)

Motivated by theoretical problems in deep learning, we conjecture that post-composing a fixed number of pairwise distinct nonconstant polynomials with a generic polynomial of sufficiently large degree yields linearly independent polynomials. This generalizes Newman--Slater's theorem on powers of polynomials. We establish several cases of this conjecture and its origin-passing variant: We prove the result for two polynomials, and for an arbitrary number of polynomials when their degrees are bounded. Furthermore, we show how the conjecture implies a complete understanding of the identifiability (i.e., parameter symmetries) of deep fully connected neural network architectures with generic polynomial activation functions. In particular, for network architectures with layer-specific activations of increasing degree, our established versions of the conjecture fully characterize the set of parameters yielding the same end-to-end network function. As a special case, we fully resolve the identifiability of shallow polynomial networks.