Spectral methods include a family of algorithms related to the eigenvectors of certain data-generated matrices. In this work, we are interested in studying the geometric landscape of the eigendecomposition problem in various spectral methods. In particular, we first extend known results regarding the landscape at critical points to larger regions near the critical points in a special case of finding the leading eigenvector of a symmetric matrix. For a more general eigendecomposition problem, inspired by recent findings on the connection between the landscapes of empirical risk and population risk, we then build a novel connection between the landscape of an eigendecomposition problem that uses random measurements and the one that uses the true data matrix. We also apply our theory to a variety of low-rank matrix optimization problems and conduct a series of simulations to illustrate our theoretical findings.

翻译：光谱方法包括与某些数据生成矩阵的分解器有关的一系列算法。 在这项工作中,我们有兴趣研究各种光谱方法中的微分分分解问题的几何景观。 特别是,我们首先将关于临界点地貌的已知结果推广到靠近临界点的较大区域,这是在发现一个对称矩阵的主要分解器的特殊案例中发现一个关键点。 对于由于最近关于实证风险和人口风险地貌之间联系的调查结果而引发的更一般的微分分解问题,我们随后在使用随机测量法的微分组合问题的地貌与使用真实数据矩阵的地貌之间建立了新的联系。 我们还将我们的理论应用于各种低层次矩阵优化问题,并进行一系列模拟,以说明我们的理论发现。