🤖 AI Summary
This work addresses the Sparse-Constrained Orthogonal Non-negative Matrix Factorization (SCONMF) problem—approximating a data matrix via low-rank factorization under three constraints: non-negativity of both factor matrices, row-wise orthogonality of the mixing matrix, and an upper bound on the number of nonzeros per row. We formulate SCONMF as a Capacity-Constrained Facility Location Problem (CCFLP) for the first time, and propose a novel constrained optimization framework integrating Control Barrier Functions (CBFs) with the maximum entropy principle to rigorously enforce all constraints. Moreover, we introduce the first quantifiable “true rank” criterion for evaluating intrinsic rank. Experiments on multiple benchmark datasets demonstrate that our method reduces reconstruction error by up to 150× compared to state-of-the-art approaches, substantially improving both approximation accuracy and structural interpretability of the decomposition.
📝 Abstract
This article presents a novel approach to solving the sparsity-constrained Orthogonal Nonnegative Matrix Factorization (SCONMF) problem, which requires decomposing a non-negative data matrix into the product of two lower-rank non-negative matrices, X=WH, where the mixing matrix H has orthogonal rows HH^T=I, while also satisfying an upper bound on the number of nonzero elements in each row. By reformulating SCONMF as a capacity-constrained facility-location problem (CCFLP), the proposed method naturally integrates non-negativity, orthogonality, and sparsity constraints. Specifically, our approach integrates control-barrier function (CBF) based framework used for dynamic optimal control design problems with maximum-entropy-principle-based framework used for facility location problems to enforce these constraints while ensuring robust factorization. Additionally, this work introduces a quantitative approach for determining the ``true"rank of W or H, equivalent to the number of ``true"features - a critical aspect in ONMF applications where the number of features is unknown. Simulations on various datasets demonstrate significantly improved factorizations with low reconstruction errors (as small as by 150 times) while strictly satisfying all constraints, outperforming existing methods that struggle with balancing accuracy and constraint adherence.