论文标题
对称阳性矩阵，riemannian歧管和谎言组的加性模型
Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups
论文作者
论文摘要
在本文中，提出了对称正定基质矩阵响应和多个标量预测指标的添加回归模型。该模型利用了从Log-Cholesky公制或对数 - 欧几里得框架继承的Abelian群体结构，该框架将对称的正量准矩阵的空间转变为Riemannian歧管，并进一步二线谎言组。显示了具有这些指标中的任何一个的对称正定矩阵空间中响应的添加模型，可在切线空间上连接到添加剂模型。该连接不仅需要有效的算法来估计组件函数，而且还允许将所提出的加性模型推广到可能没有谎言组结构的一般riemannian歧管。还建立了最佳的渐近收敛速率和估计组件函数的正态性。数值研究表明，所提出的模型具有出色的数值性能，尤其是在有多个预测因子时。通过分析扩散张量脑成像数据来证明所提出模型的实际优点。
In this paper an additive regression model for a symmetric positive-definite matrix valued response and multiple scalar predictors is proposed. The model exploits the abelian group structure inherited from either the Log-Cholesky metric or the Log-Euclidean framework that turns the space of symmetric positive-definite matrices into a Riemannian manifold and further a bi-invariant Lie group. The additive model for responses in the space of symmetric positive-definite matrices with either of these metrics is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions but also allows to generalize the proposed additive model to general Riemannian manifolds that might not have a Lie group structure. Optimal asymptotic convergence rates and normality of the estimated component functions are also established. Numerical studies show that the proposed model enjoys superior numerical performance, especially when there are multiple predictors. The practical merits of the proposed model are demonstrated by analyzing diffusion tensor brain imaging data.
