# Manifold Based Low-rank Regularization for Image Restoration and Semi-supervised Learning

## TL;DR

This paper introduces a manifold-based low-rank regularization technique that effectively handles nonlinear data structures, improving performance in various image restoration and semi-supervised learning tasks.

## Contribution

It proposes a novel manifold-based low-rank regularization method that offers greater flexibility for nonlinear data, outperforming existing approaches in image and data science applications.

## Key findings

- Effective in image inpainting and super-resolution
- Improves X-ray CT image reconstruction quality
- Enhances semi-supervised learning accuracy on handwritten digits

## 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.