mlsanityData to 7 Oct 2026

The Euclidean distance degree of an algebraic variety

arXiv 1309.0049 ↗First posted 31 Aug 2013math.AG math.OCcited by 15

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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.

Key facts. Cited by 15 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 Aug 2013.

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Citing papers per year: 2021: 2; 2022: 2; 2023: 2; 2024: 3; 2025: 5; 2026: 1.

15 papers.

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  1. 2021Geometry of Linear Convolutional Networkscited by 27
  2. 2025Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometrycited by 21
  3. 2024On the Geometry and Optimization of Polynomial Convolutional Networkscited by 11
  4. 2024Algebraic Complexity and Neurovariety of Linear Convolutional Networkscited by 10
  5. 2025Geometry and Optimization of Shallow Polynomial Networkscited by 10
  6. 2021The Geometry of Memoryless Stochastic Policy Optimization in Infinite-Horizon POMDPscited by 9
  7. 2022Snapshot of Algebraic Visioncited by 9
  8. 2024Revisiting Sampson Approximations for Geometric Estimation Problemscited by 6
  9. 2022An Atlas for the Pinhole Cameracited by 5
  10. 2023Geometry of Linear Neural Networks: Equivariance and Invariance under Permutation Groupscited by 5
  11. 2026Robustness Verification of Polynomial Neural Networkscited by 3
  12. 2023Algebra and Geometry of Camera Resectioningcited by 2
3 more
  1. 2025Testing the variety hypothesiscited by 1
  2. 2025Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrizationcited by 1
  3. 2025Parametrising the Inhomogeneity Inducing Capacity of a Training Set, and its Impact on Supervised Learningcited by 0