Solving the so-called geodesic endpoint problem, i.e., finding a geodesic that connects two given points on a manifold, is at the basis of virtually all data processing operations, including averaging, clustering, interpolation and optimization. On the Stiefel manifold of orthonormal frames, this problem is computationally involved. A remedy is to use quasi-geodesics as a replacement for the Riemannian geodesics. Quasi-geodesics feature constant speed and covariant acceleration with constant (but possibly non-zero) norm. For a well-known type of quasi-geodesics, we derive a new representation that is suited for large-scale computations. Moreover, we introduce a new kind of quasi-geodesics that turns out to be much closer to the Riemannian geodesics.
翻译:解决所谓的大地学终点问题,即找到连接一个方块上两个给点的大地学,是几乎所有数据处理作业的基础,包括平均、集聚、内插和优化。在正方形框架的Stiefel 方块上,这个问题在计算上涉及。一种补救措施是使用准地球学来替代里格曼的大地学。 准地球学以恒定速度和共变加速为恒定( 但可能非零)标准。 对于一种众所周知的准地球学,我们产生一种新的代表,适合大规模计算。此外,我们引入了一种新的准地球学,结果与里格曼的大地学非常接近。