Euclidean distance degree in manifold optimization
Zehua Lai, Lek-Heng Lim, Ke Ye
TL;DR
This work computes the Euclidean distance degree (ED) for the flag, Grassmann, and Stiefel manifolds embedded as real affine subvarieties, using their complex loci to apply algebraic geometry. The authors show that EDs take explicit, parameter-free forms: $\text{ED}( ext{Flag})=\binom{n}{k_1,k_2-k_1,\dots,n-k_p}$, $\text{ED}(\text{Gr})=\binom{n}{k}$, and $\text{ED}(\text{V}_B)=2^k$, with extensions to Schubert subvarieties yielding $\binom{m-k}{l-k}$. They provide exact stationary-point characterizations and, in many cases, closed-form nearest-point (projection) solutions, shedding light on the computational complexity of manifold optimization. The results suggest inherent tractability challenges for these common manifolds while framing ED as a useful, objective complexity measure; the paper also outlines open problems on the symplectic Grassmannian and general Schubert varieties.
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.