Stiefel流形及其商流形上计算黎曼对数的闭式测地线与信赖域方法
Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds
AI总结：
本文针对Stiefel流形及其商流形提出闭式测地线公式，并结合信赖域优化平方Frobenius距离来计算黎曼对数映射与测地距离，为无闭式对数映射的流形提供了新计算框架。
摘要：
我们为Stiefel流形上的一族度量提供了两个闭式测地线公式，该度量族由两个正数参数化，并以嵌入度量和典型度量作为特例。这些闭式公式使我们能够通过降维矩阵指数来计算低秩流形上的测地线。结合使用Fr{é}chet导数来计算测地线终点与流形上给定点之间平方Frobenius距离的梯度，我们证明了流形上两个端点之间的对数映射和测地距离可以通过信赖域求解器最小化该平方距离来计算。这为具有已知测地线公式但没有闭式对数映射的流形提供了一个计算测地距离的新框架。我们展示了该方法在Stiefel流形以及旗流形上表现良好。对数映射可用于计算配备上述度量的这些流形的黎曼质心。我们还推导了Grassmann流形上黎曼指数映射和对数映射的简单三角公式。
英文摘要：
We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifold, parameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute geodesics by matrix exponential in reduced dimension for low-rank manifolds. Combining with the use of Fr{é}chet derivatives to compute the gradient of the square Frobenius distance between a geodesic ending point to a given point on the manifold, we show the logarithm map and geodesic distance between two endpoints on the manifold could be computed by {\it minimizing} this square distance by a {\it trust-region} solver. This leads to a new framework to compute the geodesic distance for manifolds with known geodesic formula but no closed-form logarithm map. We show the approach works well for Stiefel as well as flag manifolds. The logarithm map could be used to compute the Riemannian center of mass for these manifolds equipped with the above metrics. We also deduce simple trigonometric formulas for the Riemannian exponential and logarithm maps on the Grassmann manifold.
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