发表机构
Department of Statistics, Faculty of Arts and Sciences at Beijing Normal University, Zhuhai（北京师范大学珠海分校文理学院统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对谱方法在噪声等情况下性能受限问题，提出RPMA框架用于秩 - K投影矩阵鲁棒估计，将模型转化为格拉斯曼流形上的优化问题，推导相关条件和稳定性，开发高效算法，实验证明其在投影矩阵恢复及社区检测、聚类方面优于传统方法。
AI 中文摘要
谱方法是社区检测、聚类和图学习中广泛使用的技术。但其性能严重依赖于潜在谱子空间的准确估计，在噪声、异常值或模型扰动存在时会大幅下降。为解决此局限，我们提出正则化投影矩阵近似（RPMA）框架用于秩 - K投影矩阵的鲁棒估计。它通过纳入正则化项扩展经典谱投影，产生更鲁棒、稀疏且可解释的投影估计。我们将模型表述为秩 - K投影矩阵流形上的优化问题，并利用其与格拉斯曼流形的几何等价性。基于此流形特征，推导一阶和二阶最优性条件，建立正则化主导特征子空间的局部稳定性，并刻画足够小正则化下临界点景观的稳定性。为有效解决所得非凸优化问题，开发了带回溯线搜索的黎曼梯度投影算法以及更高效的避免重复特征分解的凯莱 - 谢尔曼 - 莫里森 - 伍德伯里（Cayley - SMW）梯度方法。在合成和真实世界数据集上的大量实验表明，RPMA显著提高了投影矩阵的恢复精度，在噪声环境下的社区检测和聚类中始终优于传统谱投影方法。
英文摘要
Spectral methods are among the most widely used techniques for community detection, clustering, and graph learning. Their performance, however, critically depends on the accurate estimation of the underlying spectral subspace and can deteriorate substantially in the presence of noise, outliers, or model perturbations. To address this limitation, we propose a Regularized Projection Matrix Approximation (RPMA) framework for robust estimation of rank-$K$ projection matrices. RPMA extends classical spectral projection by incorporating a regularization term, producing projection estimates that are more robust, sparse, and interpretable. We formulate the proposed model as an optimization problem on the manifold of rank-$K$ projection matrices and exploit its geometric equivalence to the Grassmann manifold. Based on this manifold characterization, we derive the first- and second-order optimality conditions, establish the local stability of the regularized leading eigenspace, and characterize the stability of the critical-point landscape under sufficiently small regularization. To efficiently solve the resulting nonconvex optimization problem, we develop a Riemannian gradient projection algorithm with backtracking line search, together with a more efficient Cayley--Sherman--Morrison--Woodbury (Cayley--SMW) gradient method that avoids repeated eigendecompositions. Extensive experiments on both synthetic and real-world datasets demonstrate that RPMA substantially improves the recovery accuracy of projection matrices and consistently outperforms conventional spectral projection methods for community detection and clustering under noisy environments.
