# The Euclidean distance degree of an algebraic variety

arXiv: https://arxiv.org/abs/1309.0049 · HTML: https://mlsanity.com/paper/1309.0049/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Aug 2013.

## Built on by (most cited first, 16 of 16)

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- [Revisiting Sampson Approximations for Geometric Estimation Problems](https://mlsanity.com/paper/2401.07114/) (2024)
- [An Atlas for the Pinhole Camera](https://mlsanity.com/paper/2206.13468/) (2022)
- [Geometry of Linear Neural Networks: Equivariance and Invariance under Permutation Groups](https://mlsanity.com/paper/2309.13736/) (2023)
- [Robustness Verification of Polynomial Neural Networks](https://mlsanity.com/paper/2602.06105/) (2026)
- [Algebra and Geometry of Camera Resectioning](https://mlsanity.com/paper/2309.04028/) (2023)
- [Testing the variety hypothesis](https://mlsanity.com/paper/2507.16705/) (2025)
- [Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization](https://mlsanity.com/paper/2512.21029/) (2025)
- [Parametrising the Inhomogeneity Inducing Capacity of a Training Set, and its Impact on Supervised Learning](https://mlsanity.com/paper/2510.18332/) (2025)
- [What Can a Gaussian Process Design Test](https://mlsanity.com/paper/2610.10122/) (2026)

## Abstract (arXiv)

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.