Constrained Nonnegative Matrix Factorization for Image Representation
Nonnegative matrix factorization (NMF) is a popular technique for finding parts-based, linear representations of nonnegative data. It has been successfully applied in a wide range of applications such as pattern recognition, information retrieval, and computer vision. However, NMF is essentially an unsupervised method and cannot make use of label information. In this paper, we propose a novel semi-supervised matrix decomposition method, called Constrained Nonnegative Matrix Factorization (CNMF), which incorporates the label information as additional constraints. Specifically, we show how explicitly combining label information improves the discriminating power of the resulting matrix decomposition. We explore the proposed CNMF method with two cost function formulations and provide the corresponding update solutions for the optimization problems. Empirical experiments demonstrate the effectiveness of our novel algorithm in comparison to the state-of-the-art approaches through a set of evaluations based on real-world applications.
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.
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