# Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization

arXiv: https://arxiv.org/abs/2512.21029 · HTML: https://mlsanity.com/paper/2512.21029/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 27 references that are in the map.
- First posted on arXiv 24 Dec 2025.

## Builds on (most cited first, 10 of 10)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021)
- [Pure and Spurious Critical Points: a Geometric Study of Linear Networks](https://mlsanity.com/paper/1910.01671/) (2019)
- [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)
- [The Euclidean distance degree of an algebraic variety](https://mlsanity.com/paper/1309.0049/) (2013)
- [Geometry of Lightning Self-Attention: Identifiability and Dimension](https://mlsanity.com/paper/2408.17221/) (2024)
- [On the Geometry and Optimization of Polynomial Convolutional Networks](https://mlsanity.com/paper/2410.00722/) (2024)
- [Algebraic Complexity and Neurovariety of Linear Convolutional Networks](https://mlsanity.com/paper/2401.16613/) (2024)
- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (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 study the critical points over an algebraic variety of an optimization problem defined by a quadratic objective that is degenerate. This scenario arises in machine learning when the dataset size is small with respect to the model, and is typically referred to as overparametrization. Our main result relates the degenerate optimization problem to a nondegenerate one via a projection. In the highly-degenerate regime, we find that a central role is played by the ramification locus of the projection. Additionally, we provide tools for counting the number of critical points over projective varieties, and discuss specific cases arising from deep learning. Our work bridges tools from algebraic geometry with ideas from machine learning, and it extends the line of literature around the Euclidean distance degree to the degenerate setting.