论文标题
对称特征值问题的地球凸性和Riemannian陡峭下降的收敛性
Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Riemannian Steepest Descent
论文作者
论文摘要
我们研究了riemannian最陡峭的下降算法的融合，以最大程度地减少对称矩阵的雷利商的块版。即使从欧几里得意义上讲，这个问题是非凸的，并且在里曼尼亚的意义上只有本地凸，但我们发现了这个问题的结构，类似于地球强的凸性，即弱凸性。这使我们可以在研究最陡峭的下降算法的收敛时应用凸优化的类似参数，但初始化条件不取决于eigengap $δ$。当$δ> 0 $时，我们证明指数收敛速率，而否则收敛为代数。此外，我们证明此问题是在半径$ \ MATHCAL {O}（\sqrtδ）$的全局最小化器的邻域中凸出的。
We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense and only very locally convex in the Riemannian sense, we discover a structure for this problem that is similar to geodesic strong convexity, namely, weak-strong convexity. This allows us to apply similar arguments from convex optimization when studying the convergence of the steepest descent algorithm but with initialization conditions that do not depend on the eigengap $δ$. When $δ>0$, we prove exponential convergence rates, while otherwise the convergence is algebraic. Additionally, we prove that this problem is geodesically convex in a neighbourhood of the global minimizer of radius $\mathcal{O}(\sqrtδ)$.
