Schubert变种与不同维度子空间间距离的计算
Schubert varieties and distances between subspaces of different dimensions
AI总结：
研究如何扩展子空间距离以处理不同维度的子空间，提出通过Schubert变种到Grassmannian的距离方法，具有明确表达和高效计算特性。
摘要：
我们解决了一个在应用中经常出现的基本问题：如何将通常的Grassmann距离在等维子空间之间扩展到不同维度的子空间？我们展示了一个自然的解决方案是通过Grassmannian内的Schubert变种到点的距离。这种距离在子空间等维时退化为Grassmann距离，并不依赖于任何嵌入到更大环境空间。此外，它具有涉及主角的明确表达，并且可以以数值稳定的方式高效计算。我们的结果在很大程度上独立于Grassmann距离---如果需要，可以替换为任何其他常见的子空间距离。我们的方法依赖于Grassmannian的明确代数几何观点，这与在应用和计算数学中广泛建立的微分几何观点相平行。
英文摘要：
We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance --- if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well-established in applied and computational mathematics.
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