论文标题
大地回归
Geodesic regression
论文作者
论文摘要
地球回归理论旨在找到一条地质曲线，这是对给定数据集的最佳拟合。在本文中，我们将自己限制在有限维度的希尔伯特（Hilbert）空间上的正面确定运营商（矩阵）的里曼尼亚人歧管。有一条独特的地球曲线，连接两个正定算子，并且由加权几何平均值给出。测量操作员和测量曲线之间平方的riemannian度量距离的功能不是凸的，也不是在运算符中生成曲线的凸面。这与线性回归的情况有明显的区别。文献主要试图找到在单个点上近似最佳曲线的数值解。 我们建议采用比里曼尼亚度量的距离稍微擦拭的距离。随后的控制功能忠实地识别了测地曲线，并且它与基于riemannian通勤操作员的标准控制函数一致。以这种方式构建的控制函数是地理凸。因此，我们能够找到对任何给定数据集的全局且独特定义的最佳拟合。测量曲线的发电机也可以确定为两个操作员方程的唯一解决方案。
The theory of geodesic regression aims to find a geodesic curve which is an optimal fit to a given set of data. In this article we restrict ourselves to the Riemannian manifold of positive definite operators (matrices) on a Hilbert space of finite dimension. There is a unique geodesic curve connecting two positive definite operators, and it is given by the weighted geometric mean. The function that measures the squared Riemannian metric distance between an operator and a geodesic curve is not convex nor geodesically convex in the operators generating the curve. This is a marked difference to the situation in linear regression. The literature mainly tries to find numerical solutions that approximate the optimal curve in a single point. We suggest to apply a distance measure slightly coarser than the Riemannian metric. The ensuing control function faithfully identifies geodesic curves, and it coincides with the standard control function based on the Riemannian metric for commuting operators. The control function constructed in this way is geodesically convex. We are therefore able to find a global and uniquely defined optimal fit to any given set of data. The generators of the geodesic curve may also be determined as the unique solution to two operator equations.
