Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Steepest Descent
研究了Grassmann流形上瑞利商块版本的黎曼最速下降算法收敛性，发现弱强凸性结构，证明当特征间隙大于0时指数收敛，否则代数收敛。
Abstract 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 $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> . When $$\delta >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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 $${\mathcal {O}}(\sqrt{\delta })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mi>δ</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .