# CURE: Curvature Regularization For Missing Data Recovery

## TL;DR

This paper introduces CURE, a novel regularization combining low-dimensional manifold constraints with curvature smoothness, improving missing data recovery in imaging tasks.

## Contribution

The paper proposes CURE, a new regularization method that integrates manifold low dimension and curvature smoothness, enhancing image inpainting and semi-supervised learning.

## Key findings

- CURE outperforms LDMM in image inpainting.
- WeCURE improves semi-supervised learning results.
- Numerical experiments validate the effectiveness of the proposed methods.

## 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.