论文标题
在自我符合函数的多种表述中
On a manifold formulation of self-concordant functions
论文作者
论文摘要
在本文中，我们介绍了一种自我符合函数理论的延伸。我们通过其Hessian的协方差衍生物的条件来制定地理上凸功能的自我符号，并验证许多类似特性（例如牛顿方法的二次收敛）和路径遵循方法的多项式迭代复杂性的二次收敛。但是，尚不清楚非欧国人流形是否真的存在有用的自我符合功能/障碍。在这个问题上，我们提供了一个初步的结果，即曲率$-κ$的双曲线空间中的平方距离函数为$ \sqrtκ/2 $ -sself-concordant和Radius $ r $的相关对数屏障$ r $是$ O（κR^2）$ - 自我控制障碍。我们还向双曲线空间中的最小封闭球提出了应用。
In this paper, we address an extension of the theory of self-concordant functions for a manifold. We formulate the self-concordance of a geodesically convex function by a condition of the covariant derivative of its Hessian, and verify that many of the analogous properties, such as the quadratic convergence of Newton's method and the polynomial iteration complexity of the path-following method, are naturally extended. However it is not known whether a useful class of self-concordant functions/barriers really exists for non-Euclidean manifolds. To this question, we provide a preliminary result that the squared distance function in the hyperbolic space of curvature $- κ$ is $\sqrtκ/2$-self-concordant and the associated logarithmic barrier of a ball of radius $R$ is an $O(κR^2)$-self-concordant barrier. We also give an application to the minimum enclosing ball in a hyperbolic space.
