Community discovery is a crucial issue in the realm of complex network analysis. A Symmetric Non-negative Matrix Factorization (SNMF)-based method is widely used to address this problem. It utilizes a single feature matrix to capture network symmetry, which limits its ability to learn node representations. To break this limitation, this study introduces a novel Relaxed Symmetric Non-negative Matrix Factorization (RSN) approach to enhance an SNMF-based community detector. It works by a) expanding the representational space and its degrees of freedom relying on multiple feature matrices; b) integrating the well-designed equality constraints to enable the model to perceive the network’s intrinsic symmetry better; c) employing graph regularization to maintain the local geometric invariance of the network structure; and d) separating constraints from decision variables for efficient optimization via the alternating-direction-method of multipliers (ADMM) principle. The RSN model’s effectiveness is demonstrated through empirical research on six social networks, showcasing superior precision in community discovery over existing models and baselines.