有界几何流形上Optimization的曲率依赖整体收敛速度
Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry
摘要：
我们给出了定义在1-有界几何流形上的弱凸函数的最优化问题的曲率依赖的收敛速度，通过黎曼梯度下降和动态平凡化算法.为了做到这一点，我们给出了一个更严格的限制的范数的海森的黎曼指数比以前已知的。我们计算这些边界显式的一些流形中常用的优化文献，如特殊的正交群和真正的格拉斯曼。一路上，我们提出了自包含的证明完全一般的指数映射的微分范数和某些余弦不等式的流形上，这是常用的流形上的优化。
英文摘要：
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.
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