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.

翻译：我们研究由退化二次目标函数定义的优化问题在代数簇上的临界点。当数据集规模相对于模型较小时，这种情形在机器学习中常出现，通常被称为过参数化。我们的主要结果通过投影将退化优化问题与非退化问题联系起来。在高退化区域中，我们发现投影的分支轨迹起着核心作用。此外，我们提供了计算射影簇上临界点数量的工具，并讨论了深度学习中的具体案例。本研究将代数几何工具与机器学习思想相结合，并将欧氏距离度相关文献延伸至退化情形。