Fuzzy Multi-Subspace Clustering

Projection-based subspace clustering methods typically learn a single projection subspace shared across all clusters. This strategy prioritizes consistency across clusters at the expense of their distinctiveness—potentially misaligning the projection objective with the clustering goal. To overcome this limitation, we propose Fuzzy Multi-Subspace Clustering (FMSC), which simultaneously learns multiple cluster-specific subspaces and membership assignments in a mutually reinforcing manner. The refinement of subspaces helps improve membership accuracy, while more accurate membership assignments guide more precise subspace learning. Moreover, incorporating fuzzy learning enables FMSC to perform one-step clustering and handle scenarios in which a sample simultaneously belongs to multiple subspaces. We also propose a novel subspace regularization method for FMSC that maintains cross-subspace consistency without sacrificing the fidelity of cluster-specific subspace reconstruction. The time complexity of FMSC scales linearly with the sample size, ensuring its scalability. Extensive experiments on synthetic datasets with millions of samples and widely used benchmarks demonstrate that FMSC outperforms several state-of-the-art methods.