发表机构
University of Hertfordshire; Sefako Makgatho Health Sciences University（赫特福德大学; 塞法科·马卡索健康科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出RIAG-R，一种结合自适应步长、惯性外推和梯度触发重启的黎曼优化方法，在多个矩阵流形上显著优于基线，证明了自适应重启对曲率空间动量的关键作用。
AI 中文摘要
我们提出RIAG-R，一种带梯度触发重启的黎曼惯性自适应梯度方法，用于黎曼流形上的优化。RIAG-R结合了三个要素：一种AdaGrad型自适应步长，无需知道Lipschitz常数；通过指数映射在切空间中进行惯性外推以加速收敛；以及一种每次迭代仅需额外一次内积的重启规则，用于检测并纠正由流形曲率引起的动量饱和病理。我们建立了每次迭代的下降不等式、最优的$O(\varepsilon^{-2})$迭代复杂度、生成序列的聚点收敛结果，以及在黎曼Polyak-Łojasiewicz不等式下的线性收敛速率。在四个代表性矩阵流形优化问题（包括球面、Stiefel、对称正定和Grassmann流形）上的数值实验表明，RIAG-R始终优于非惯性和无重启的惯性基线，证实了自适应重启对于在弯曲空间上实现动量优势至关重要。
英文摘要
We propose RIAG-R, a Riemannian Inertial Adaptive Gradient method with gradient-triggered Restart, for optimization on Riemannian manifolds. RIAG-R combines three ingredients: an AdaGrad-type adaptive step-size that requires no knowledge of the Lipschitz constant; inertial extrapolation in the tangent space via the exponential map to accelerate convergence; and a restart rule, costing only one extra inner product per iteration, that detects and corrects the momentum-saturation pathology induced by manifold curvature. We establish a per-iteration descent inequality, an optimal $O(\varepsilon^{-2})$-iteration complexity, a convergence result for cluster points of the generated sequence, and a linear convergence rate under the Riemannian Polyak-Łojasiewicz inequality. Numerical experiments on four representative matrix manifold optimization problems including sphere, Stiefel, symmetric positive-definite, and Grassmann manifolds demonstrate that RIAG-R consistently outperforms non-inertial and restart-free inertial baselines, confirming that adaptive restart is essential to realizing the benefits of momentum on curved spaces.
