论文标题
牛顿缩回作为子手术的近似大地测量学
Newton retraction as approximate geodesics on submanifolds
论文作者
论文摘要
大地学的有效近似对于流形的实用算法至关重要。在这里，我们引入了一类关于子曼群的缩回，这是由环境歧管的叶子引起的。它们将投影缩回与三阶匹配，从而将指数映射与第二阶匹配。特别是，我们表明，牛顿缩回（NR）始终比流行的方法被称为倾斜投影或拼字法回收：每个Kantorovich型收敛定理，NR的超线性收敛区域包括后者的超线性收敛区域。我们还表明，NR始终具有较低的计算成本。 NR的优选属性可用于优化，采样和许多其他统计问题。
Efficient approximation of geodesics is crucial for practical algorithms on manifolds. Here we introduce a class of retractions on submanifolds, induced by a foliation of the ambient manifold. They match the projective retraction to the third order and thus match the exponential map to the second order. In particular, we show that Newton retraction (NR) is always stabler than the popular approach known as oblique projection or orthographic retraction: per Kantorovich-type convergence theorems, the superlinear convergence regions of NR include those of the latter. We also show that NR always has a lower computational cost. The preferable properties of NR are useful for optimization, sampling, and many other statistical problems on manifolds.
