Euclidean distance degree in manifold optimization

2025/02/14 by Zehua Lai, Lek‐Heng Lim, Lai, Zehua +3 · 2 citations
Engineering · #14M15 #15A24 #58C05 #58K05 #90C26 #Advanced Measurement and Metrology Techniques #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2502.10336

openalex publication_date 2025/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine the Euclidean distance degrees of the three most common manifolds arising in manifold optimization: flag, Grassmann, and Stiefel manifolds. For the Grassmannian, we will also determine the Euclidean distance degree of an important class of Schubert varieties that often appear in applications. Our technique goes further than furnishing the value of the Euclidean distance degree; it will also yield closed-form expressions for all stationary points of the Euclidean distance function in each instance. We will discuss the implications of these results on the tractability of manifold optimization problems.

Cited by

• The Euclidean distance degree of curves: from rational to line multiview varieties
• Grassmann and Flag Varieties in Linear Algebra, Optimization, and Statistics: An Algebraic Perspective

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• The Euclidean distance degree of an algebraic variety
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