Symmetric matrix factorization (SMF) plays an important role in clustering and representation learning. Nevertheless, most existing SMF-based approaches are formulated as non-convex optimization problems, which often leads to unstable convergence and high computational costs. In this paper, we develop a fast unconstrained convex symmetric matrix factorization framework, termed FUCSMF, for semi-supervised learning. By incorporating label information into the symmetric factorization formulation, the proposed model is transformed into a convex objective, which guarantees global optimality and enables efficient optimization using standard unconstrained solvers. To further improve scalability, a bipartite graph structure is introduced into SMF from a hypergraph-inspired perspective, significantly reducing the computational burden. The resulting computational complexity is reduced to $O\left(nmd\right)$, which is substantially lower than the $O(nmd+m^{2}n+m^3)$ complexity required by existing bipartite graph-based methods, where n, m, and d denote the numbers of samples, anchor points, and feature dimensions, respectively. In addition, we propose a correntropy-based graph construction strategy to alleviate the sensitivity of conventional adaptive neighbor bipartite graph methods. Extensive experiments on six benchmark datasets, involving comparisons with eleven state-of-the-art methods, demonstrate that FUCSMF achieves superior clustering performance while requiring significantly less computational time. Empirical results further show that the proposed method converges rapidly, typically within ten iterations.

Matrix factorization has demonstrated outstanding performance in machine learning. Recently, graph-based matrix factorization has gained widespread attention. However, graph-based methods are only suitable for handling small amounts of data. This paper proposes a fast semi-supervised learning method using only matrix factorization, which considers both global and local information. By introducing bipartite graphs into symmetric matrix factorization, the technique can handle large datasets effectively. It is worth noting that by utilizing tag information, the proposed symmetric matrix factorization becomes convex and unconstrained, i.e., the non-convex problem ${min}_x{(1-x^2)}^2$ is transformed into a convex problem. This allows it to be optimized quickly using state-of-the-art unconstrained optimization algorithms. The computational complexity of the proposed method is $O\left(nmd\right)$, which is much lower than that of the original symmetric matrix factorization, which is $O\left(n^{2}d\right)$, and even lower than that of other anchor-based methods, which is $O(nmd+m^{2}n+m^3)$, where n represents the number of samples, d represents the number of features, and $m\ll n$ represents the number of anchors. The experimental results on multiple public datasets indicate that the proposed method achieves higher performance in less time.