Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization
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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.
Key facts. Cited by 1 paper 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
10 of 27 references.
References · 10
- 2017Attention Is All You Needcited by 60,361
- 2021Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolationcited by 132
- 2019Pure and Spurious Critical Points: a Geometric Study of Linear Networkscited by 33
- 2024Geometry of Polynomial Neural Networkscited by 21
- 2025Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometrycited by 21
- 2013The Euclidean distance degree of an algebraic varietycited by 16
- 2024Geometry of Lightning Self-Attention: Identifiability and Dimensioncited by 16
- 2024On the Geometry and Optimization of Polynomial Convolutional Networkscited by 11
- 2024Algebraic Complexity and Neurovariety of Linear Convolutional Networkscited by 10
- 2025Geometry and Optimization of Shallow Polynomial Networkscited by 10
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Citing papers per year: 2026: 1.
1 paper.