Robust Structured Nonnegative Matrix Factorization for Image Representation
Dimensionality reduction has attracted increasing attention, because high-dimensional data have arisen naturally in numerous domains in recent years. As one popular dimensionality reduction method, nonnegative matrix factorization (NMF), whose goal is to learn parts-based representations, has been widely studied and applied to various applications. In contrast to the previous approaches, this paper proposes a novel semisupervised NMF learning framework, called robust structured NMF, that learns a robust discriminative representation by leveraging the block-diagonal structure and the <inline-formula> <tex-math notation="LaTeX">$\ell _{2,p}$ </tex-math></inline-formula>-norm (especially when <inline-formula> <tex-math notation="LaTeX">$0<p\leq 1$ </tex-math></inline-formula>) loss function. Specifically, the problems of noise and outliers are well addressed by the <inline-formula> <tex-math notation="LaTeX">$\ell _{2,p}$ </tex-math></inline-formula>-norm (<inline-formula> <tex-math notation="LaTeX">$0<p\leq 1$ </tex-math></inline-formula>) loss function, while the discriminative representations of both the labeled and unlabeled data are simultaneously learned by explicitly exploring the block-diagonal structure. The proposed problem is formulated as an optimization problem with a well-defined objective function solved by the proposed iterative algorithm. The convergence of the proposed optimization algorithm is analyzed both theoretically and empirically. In addition, we also discuss the relationships between the proposed method and some previous methods. Extensive experiments on both the synthetic and real-world data sets are conducted, and the experimental results demonstrate the effectiveness of the proposed method in comparison to the state-of-the-art methods.