# Manifold Graph Signal Restoration using Gradient Graph Laplacian Regularizer

arXiv: https://arxiv.org/abs/2206.04245 · HTML: https://mlsanity.com/paper/2206.04245/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2022.

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

- [Interpretable Lightweight Transformer via Unrolling of Learned Graph Smoothness Priors](https://mlsanity.com/paper/2406.04090/) (2024)
- [Efficient Signed Graph Sampling via Balancing & Gershgorin Disc Perfect Alignment](https://mlsanity.com/paper/2208.08726/) (2022)
- [Efficient Learning of Balanced Signed Graphs via Iterative Linear Programming](https://mlsanity.com/paper/2409.07794/) (2024)
- [Retinex-based Image Denoising / Contrast Enhancement using Gradient Graph Laplacian Regularizer](https://mlsanity.com/paper/2307.02625/) (2023)
- [Learning to Optimize by Differentiable Programming](https://mlsanity.com/paper/2601.16510/) (2026)
- [Reconstruction of Graph Signals on Complex Manifolds with Kernel Methods](https://mlsanity.com/paper/2505.15202/) (2025)
- [Low-rank Updates in Slowly Time-varying Graphs for Spatial-Temporal Signal Interpolation](https://mlsanity.com/paper/2606.24011/) (2026)

## Abstract (arXiv)

In the graph signal processing (GSP) literature, graph Laplacian regularizer (GLR) was used for signal restoration to promote piecewise smooth / constant reconstruction with respect to an underlying graph. However, for signals slowly varying across graph kernels, GLR suffers from an undesirable "staircase" effect. In this paper, focusing on manifold graphs -- collections of uniform discrete samples on low-dimensional continuous manifolds -- we generalize GLR to gradient graph Laplacian regularizer (GGLR) that promotes planar / piecewise planar (PWP) signal reconstruction. Specifically, for a graph endowed with sampling coordinates (e.g., 2D images, 3D point clouds), we first define a gradient operator, using which we construct a gradient graph for nodes' gradients in sampling manifold space. This maps to a gradient-induced nodal graph (GNG) and a positive semi-definite (PSD) Laplacian matrix with planar signals as the 0 frequencies. For manifold graphs without explicit sampling coordinates, we propose a graph embedding method to obtain node coordinates via fast eigenvector computation. We derive the means-square-error minimizing weight parameter for GGLR efficiently, trading off bias and variance of the signal estimate. Experimental results show that GGLR outperformed previous graph signal priors like GLR and graph total variation (GTV) in a range of graph signal restoration tasks.