论文标题
依赖曲率的全局收敛速率，以优化有界几何的流形
Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry
论文作者
论文摘要
我们给出曲率依赖的收敛速率，以优化通过Riemannian梯度下降和通过动态琐事算法的1结合几何形状的歧管定义的弱凸功能。为了做到这一点，我们比以前已知的riemannian指数的黑森州的标准更加限制。我们针对优化文献中常用的一些流形明确计算这些界限，例如特殊的正交群和真实的司法群。在此过程中，我们在指数图的差异和一定余弦的差异上介绍了完全通用界限的独立范围，而这些证据通常用于歧管的优化。
We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization algorithm. In order to do this, we give a tighter bound on the norm of the Hessian of the Riemannian exponential than the previously known. We compute these bounds explicitly for some manifolds commonly used in the optimization literature such as the special orthogonal group and the real Grassmannian. Along the way, we present self-contained proofs of fully general bounds on the norm of the differential of the exponential map and certain cosine inequalities on manifolds, which are commonly used in optimization on manifolds.
