🤖 AI Summary
This work addresses the sensitivity of classical spectral methods to noise, outliers, and model perturbations, which often leads to inaccurate estimation of low-rank projection subspaces and consequently degrades community detection and clustering performance. To overcome this limitation, the authors propose the Regularized Projection Matrix Approximation (RPMA) framework, formulating robust low-rank projection estimation as a regularized optimization problem on the Grassmann manifold. They establish, for the first time, first- and second-order optimality conditions for this problem and prove the local stability of the regularized dominant subspace. Furthermore, they introduce the Cayley-SMW gradient method, which leverages the Sherman–Morrison–Woodbury formula to circumvent repeated eigendecompositions, substantially improving computational efficiency. Experiments demonstrate that RPMA consistently outperforms existing spectral methods on both synthetic and real-world datasets, achieving more accurate projection recovery and enhanced clustering robustness in noisy environments.
📝 Abstract
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.