Title: Manifold Based Low-rank Regularization for Image Restoration and Semi-supervised Learning,
Abstract: Low-rank structures play important role 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 (CT) 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.