发表机构
Sony Computer Science Laboratories Inc.（索尼计算机科学实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究黎曼中点何时为Gauss复合均值，给出判据并推广Nakamura迭代至一族矩阵迭代，其Gauss复合仍为几何矩阵均值。
AI 中文摘要
Nakamura [J. Comput. Appl. Math. 131 (2001)] 证明了算术-调和矩阵迭代二次收敛于几何矩阵均值，即正定矩阵上仿射不变度量的黎曼中点。我们更一般地提出：何时黎曼中点是Gauss复合均值？一个黎曼度量和一个仿射联络诱导三个中点映射，分别对应于该联络、其度量对偶联络以及Levi-Civita联络。前两者的Gauss复合恰好等于Levi-Civita中点，当且仅当后者在迭代一步下不变；而每当一个等距变换充当关于Levi-Civita中点的点反射并交换联络与其对偶时，该不变性成立；对于足够接近的初始对，这蕴含迭代的二次收敛。Nakamura迭代是模型情形，该判据将其推广到一族矩阵迭代，其Gauss复合仍为几何矩阵均值。一个对偶平坦Hessian度量表明仅对偶性并不充分，而具有平行三次型的欧氏度量在每一大于一的维数中提供了非平坦例子。在一维情形，不变性通过Matkowski-Suto方程完全刻画，我们猜想对于欧氏对偶对，它迫使三次型为常数，我们在一维情形及常数三次型的标量倍数情形证明了这一点。
英文摘要
Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic--harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We consider the more general problem of when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection determine three midpoint maps associated with the connection, its metric dual, and the Levi--Civita connection. We show that the Gauss composition of the first two equals the Levi--Civita midpoint if and only if the latter is invariant under one step of the iteration. We give a sufficient condition for this invariance in terms of an isometry which acts as the point reflection about the Levi--Civita midpoint and exchanges the connection with its dual. Under this invariance, the iterations converge quadratically for sufficiently close initial pairs. Nakamura's iteration is an example of this criterion. The criterion also applies to dual pairs of matrix power means whose Gauss composition is the geometric matrix mean. We give a dually flat Hessian example which shows that duality alone does not suffice. We also give non-flat examples with a Euclidean metric and a parallel cubic form in every dimension larger than one. In dimension one, we characterize the invariance completely by using the Matkowski--Sutô equation. For Euclidean dual pairs, we conjecture that the invariance implies that the cubic form is constant. We prove this conjecture in dimension one and for scalar multiples of a constant cubic form.
Comments47 pages, 1 figure. Sections 3 and 5 revised
