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Manifold Based Low-Rank Regularization for Image Restoration and Semi-Supervised Learning

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Low-rank structures play important roles in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold structure has been considered in many data processing problems. Inspired by this concept, we consider a manifold based low-rank regularization as a linear approximation of manifold dimension. This regularization is less restricted than the global low-rank regularization, and thus enjoy more flexibility to handle data with nonlinear structures. As applications, we demonstrate the proposed regularization to classical inverse problems in image sciences and data sciences including image inpainting, image super-resolution, X-ray computer tomography image reconstruction and semi-supervised learning. We conduct intensive numerical experiments in several image restoration problems and a semi-supervised learning problem of classifying handwritten digits using the MINST data. Our numerical tests demonstrate the effectiveness of the proposed methods and illustrate that the new regularization methods produce outstanding results by comparing with many existing methods.

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We thank Prof. Stanley Osher, Prof. Zuoqiang Shi and Mr. Wei Zhu kindly share their valuable comments and code of both LDMM and LDMM+WGL for comparisons.

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Correspondence to Rongjie Lai.

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The research of Rongjie Lai is partially supported by NSF Grant DMS–1522645.

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Lai, R., Li, J. Manifold Based Low-Rank Regularization for Image Restoration and Semi-Supervised Learning. J Sci Comput 74, 1241–1263 (2018).

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