Kladder-NMF: Boosting NMF-based community detection with bidirectional KL-divergence reconstruction and inter-layer consistency

Community detection is a crucial task in complex network analysis that aims to find optimal cluster structures and identify valuable information. While Non-negative Matrix Factorization (NMF)-based algorithms have been widely used in this domain, most existing methods rely heavily on Frobenius norm-based objectives and adopt shallow or unidirectional reconstruction schemes. Such limitations hinder their ability to capture deep organizational patterns and structural information in complex networks. Furthermore, the information consistency between hierarchical structures is completely overlooked. To address these challenges, this paper proposes Kladder-NMF, a novel multilayer non-negative matrix factorization method. The approach constructs a second-order proximity matrix as the reconstruction target to obtain richer latent representations. It adopts KL divergence for reconstruction loss, which is more suitable for non-negative real values than traditional metrics. The method builds on a bidirectional reconstruction framework inspired by autoencoder architectures. It further introduces a novel bidirectional layer consistency term that facilitates information alignment between the top-down and bottom-up reconstruction paths. An efficient update algorithm is developed that decomposes the overall optimization problem into three subproblems. Each subproblem is solved alternately under non-negativity constraints using update rules derived from the Majorization–Minimization (MM) principle, ensuring convergence to local optima. Kladder-NMF is evaluated on multiple benchmark datasets using ARI and NMI metrics, demonstrating its superiority over existing community detection methods. Extensive ablation studies confirm the effectiveness of each proposed component.