# CURE: Curvature Regularization For Missing Data Recovery

arXiv: https://arxiv.org/abs/1901.09548 · HTML: https://mlsanity.com/paper/1901.09548/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2019.

## Built on by (most cited first, 7 of 7)

- [Regularize implicit neural representation by itself](https://mlsanity.com/paper/2303.15484/) (2023)
- [Fast Multi-grid Methods for Minimizing Curvature Energy](https://mlsanity.com/paper/2204.07921/) (2022)
- [Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions](https://mlsanity.com/paper/2307.02478/) (2023)
- [Curvature regularization for Non-line-of-sight Imaging from Under-sampled Data](https://mlsanity.com/paper/2301.00406/) (2023)
- [Continuum limit of p-biharmonic equations on graphs](https://mlsanity.com/paper/2404.19689/) (2024)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)
- [Color image restoration based on nonlocal saturation-value similarity](https://mlsanity.com/paper/2603.18586/) (2026)

## Abstract (arXiv)

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.