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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 semi supervised NMF learning framework, called robust structured NMF, that learns a robust discriminative representation by leveraging the block-diagonal structure and the l _2,p -norm (especially when 0 <; p 1) loss function. Specifically, the problems of noise and outliers are well addressed by the l _2,p -norm (0 <; p 1) loss function, while the discriminative representations of both the labeled and unlabeled data are simultaneously learned by explicitly exploring the block-diagonal structure. The …