AI 中文总结
本文研究 Grassmannian 子簇上的欧几里得距离优化问题，定义并计算其 ED 度与 GD 度，并给出若干几何子簇的公式。
AI 中文摘要
给定 Grassmannian 的一个子簇和一个数据点，我们寻求在子簇上找到使该数据点的欧几里得距离最小化的点。该优化问题的复临界点数目即为欧几里得距离（ED）度。我们证明了整个 Grassmannian 的 ED 判别式，即具有与 ED 度不同数目的临界点的数据点集合，是数据作为投影矩阵的特征多项式的判别式。另一个与 Grassmannian 中子簇紧密相关的代数复杂度度量是 Grassmann 距离（GD）度，即当数据点本身位于 Grassmannian 中时，距离优化问题的复临界点数目。我们给出了 Grassmannian 中具有几何意义的子簇的 ED 度和 GD 度的公式，即较小 Grassmannian 的乘积、拟阵实现簇和 Schubert 簇。
英文摘要
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.
Comments33 pages, 1 figure, 2 tables
