Low-rank structures play important role in recent advances of many problemsin image science and data science. As a natural extension of low-rankstructures for data with nonlinear structures, the concept of thelow-dimensional manifold structure has been considered in many data processingproblems. Inspired by this concept, we consider a manifold based low-rankregularization as a linear approximation of manifold dimension. Thisregularization is less restricted than the global low-rank regularization, andthus enjoy more flexibility to handle data with nonlinear structures. Asapplications, we demonstrate the proposed regularization to classical inverseproblems in image sciences and data sciences including image inpainting, imagesuper-resolution, X-ray computer tomography (CT) image reconstruction andsemi-supervised learning. We conduct intensive numerical experiments in severalimage restoration problems and a semi-supervised learning problem ofclassifying handwritten digits using the MINST data. Our numerical testsdemonstrate the effectiveness of the proposed methods and illustrate that thenew regularization methods produce outstanding results by comparing with manyexisting methods.
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