Non-negative patch alignment framework.

In this paper, we present a non-negative patch alignment framework (NPAF) to unify popular non-negative matrix factorization (NMF) related dimension reduction algorithms. It offers a new viewpoint to better understand the common property of different NMF algorithms. Although multiplicative update rule (MUR) can solve NPAF and is easy to implement, it converges slowly. Thus, we propose a fast gradient descent (FGD) to overcome the aforementioned problem. FGD uses the Newton method to search the optimal step size, and thus converges faster than MUR. Experiments on synthetic and real-world datasets confirm the efficiency of FGD compared with MUR for optimizing NPAF. Based on NPAF, we develop non-negative discriminative locality alignment (NDLA). Experiments on face image and handwritten datasets suggest the effectiveness of NDLA in classification tasks and its robustness to image occlusions, compared with representative NMF-related dimension reduction algorithms.