Conic Hull Fitting-Based Dictionary Matrix Learning for Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a powerful tool for signal processing and machine learning. Geometrically, it can be interpreted as the problem of finding a conic hull, which contains a cloud of data points and is embedded in the positive orthant. The separability assumption posits that the conic hull can be spanned by a small subset of the columns of the input data matrix. This assumption is equivalent to the 1-sparse condition. Many extreme-rays-based NMF methods are essentially based on the 1-sparse condition. However, the separability assumption or 1-sparse condition may not always be guaranteed for real applications. By analyzing the mathematical connection between the extreme-rays representation and the half-hyperplanes representation of a conic hull, we propose three novel NMF algorithms (i.e., HICHF, EnhancedHICHF, and ExtendedHICHF) based on the half-hyperplane identification. These algorithms can be efficiently implemented via eigenvalue decomposition (EVD). In contrast to the conventional extreme-rays-based NMF methods, the proposed methods can achieve better performance for the nonseparable NMF problems, where the 1-sparse condition is not well satisfied. Furthermore, the proposed algorithms are simple, yet efficient and more robust. Experiments on both synthetic data and real-world parts-based learning data, such as hyperspectral unmixing and facial parts learning, verify that the proposed algorithms considerably outperform the state-of-the-art algorithms.