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Euclidean Distance Optimization Within the Grassmannian

  • Hannah Friedman
  • Serkan Hoşten
  • Andrea Rosana
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Abstract

Given a subvariety of the Grassmannian and a data point, we seek to find a point on the subvariety minimizing the Euclidean distance to the data point. The number of complex critical points of this optimization problem is the Euclidean distance (ED) degree. We show that the ED discriminant of the whole Grassmannian, that is, the set of data points with a number of critical points different than the ED degree, is the discriminant of the characteristic polynomial of the data as a projection matrix. Another closely connected algebraic complexity measure for a subvariety in the Grassmannian is the Grassmann distance (GD) degree, which is the number of complex critical points of the distance optimization problem when the data point itself is in the Grassmannian. We give formulae for ED and GD degrees of geometrically meaningful subvarieties of the Grassmannian, namely, products of smaller Grassmannians, matroid realization varieties, and Schubert varieties.

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Cite

BibTeXRIS
BibTeX
@misc{friedman2026euclidean,
  title = {{Euclidean Distance Optimization Within the Grassmannian}},
  author = {Hannah Friedman and Serkan Hoşten and Andrea Rosana},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02018},
  eprint = {2610.02018},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02018},
}
RIS
TY  - UNPB
TI  - Euclidean Distance Optimization Within the Grassmannian
AU  - Hannah Friedman
AU  - Serkan Hoşten
AU  - Andrea Rosana
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.02018
UR  - https://arxiv.org/abs/2610.02018
AB  - Given a subvariety of the Grassmannian and a data point, we seek to find a point on the subvariety minimizing the Euclidean distance to the data point. The number of complex critical points of this optimization problem is the Euclidean distance (ED) degree. We show that the ED discriminant of the whole Grassmannian, that is, the set of data points with a number of critical points different than the ED degree, is the discriminant of the characteristic polynomial of the data as a projection matrix. Another closely connected algebraic complexity measure for a subvariety in the Grassmannian is the Grassmann distance (GD) degree, which is the number of complex critical points of the distance optimization problem when the data point itself is in the Grassmannian. We give formulae for ED and GD degrees of geometrically meaningful subvarieties of the Grassmannian, namely, products of smaller Grassmannians, matroid realization varieties, and Schubert varieties.
ER  -