Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912787938148352 |
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| author | Marchetti, Giovanni Luca Connelly, Erin Breiding, Paul Kohn, Kathlén |
| author_facet | Marchetti, Giovanni Luca Connelly, Erin Breiding, Paul Kohn, Kathlén |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization Marchetti, Giovanni Luca Connelly, Erin Breiding, Paul Kohn, Kathlén Algebraic Geometry Machine Learning 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. |
| title | Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization |
| topic | Algebraic Geometry Machine Learning |
| url | https://arxiv.org/abs/2512.21029 |