Title: CURE: Curvature Regularization For Missing Data Recovery,
Abstract: Missing data recovery is an important and yet challenging problem in imaging
and data science. Successful models often adopt certain carefully chosen
regularization. Recently, the low dimension manifold model (LDMM) was
introduced by S.Osher et al. and shown effective in image inpainting. They
observed that enforcing low dimensionality on image patch manifold serves as a
good image regularizer. In this paper, we observe that having only the low
dimension manifold regularization is not enough sometimes, and we need
smoothness as well. For that, we introduce a new regularization by combining
the low dimension manifold regularization with a higher order Curvature
Regularization, and we call this new regularization CURE for short. The key
step of solving CURE is to solve a biharmonic equation on a manifold. We
further introduce a weighted version of CURE, called WeCURE, in a similar
manner as the weighted nonlocal Laplacian (WNLL) method. Numerical experiments
for image inpainting and semi-supervised learning show that the proposed CURE
and WeCURE significantly outperform LDMM and WNLL respectively.