This paper is concerned with the extraction of the smallest eigenvalue and the corresponding eigenvector of a symmetric positive definite matrix pencil. We reveal implicit convexity of the eigenvalue problem in Euclidean space. A provable accelerated eigensolver based on preconditioning and implicit convexity (EPIC) is proposed. Theoretical analysis shows the acceleration of EPIC with the rate of convergence resembling the expected rate of convergence of the well-known locally optimal preconditioned conjugate gradient (LOPCG). A complete proof of the expected rate of convergence of LOPCG is elusive so far. Numerical results confirm our theoretical findings of EPIC.

翻译：本文关注对称正定矩阵束的最小特征值及对应特征向量的提取问题。我们揭示了欧氏空间中特征值问题的隐凸性，并提出了一种基于预处理与隐凸性的可证明加速特征求解器（EPIC）。理论分析表明，EPIC的加速效果具有与著名局部最优预处理共轭梯度法（LOPCG）期望收敛速率相似的收敛速率。目前尚无关于LOPCG期望收敛速率的完整证明。数值结果验证了EPIC的理论发现。