⚡ electrical engineering

Graph Signal Denoising Using Regularization by Denoising and Its Parameter Estimation

This paper proposes an interpretable graph signal denoising method based on Regularization by Denoising (RED) that leverages various graph denoisers, including graph neural networks, and introduces supervised and unsupervised parameter estimation techniques via deep algorithm unrolling to achieve superior denoising accuracy compared to existing methods.

Original authors: Hayate Kojima, Hiroshi Higashi, Yuichi Tanaka

Published 2026-05-27

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Hayate Kojima, Hiroshi Higashi, Yuichi Tanaka

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a map of a city where every building is a point, and the roads connecting them are lines. Now, imagine someone has sprayed this map with random static noise, making it hard to see the true shape of the buildings or the layout of the streets. Your goal is to clean up this map without erasing the important details.

This paper presents a new, smarter way to clean up these "noisy maps" (which the authors call graph signals). Here is how they did it, explained simply:

1. The Problem: Two Ways to Clean a Mess

Usually, people try to clean noisy data in one of two ways:

The "Rulebook" Approach (Model-Based): You follow a strict set of mathematical rules (like "neighbors should look similar"). It's very logical and easy to understand, but it can sometimes be too rigid, smoothing out important details like sharp corners or unique features.
The "Learning" Approach (Data-Driven): You train a computer brain (a neural network) on thousands of clean examples so it learns what "good" looks like. It's very powerful, but it needs a massive library of perfect examples to learn from. If you don't have those perfect examples, it struggles.

2. The Solution: "Regularization by Denoising" (RED)

The authors combined the best of both worlds using a technique called RED.

Think of RED as a smart supervisor.

Instead of just following a rigid rulebook, the supervisor hires a "cleaning expert" (a denoiser) to look at the messy map.
The supervisor asks the expert: "What would you change if you were cleaning this?"
The expert says, "I would smooth out this jagged edge here, but keep that sharp corner there."
The supervisor then uses that advice to update the map.

The magic of this paper is that they proved this "supervisor" method works not just for pictures (like cleaning up a blurry photo), but also for these complex "city maps" (graphs). They showed that many different types of "cleaning experts" (including complex AI networks) can be used in this role, provided they follow a few simple rules (like not changing the size of the map when they clean it).

3. The Secret Sauce: Learning the "Dial"

A cleaning process usually has a "dial" (a parameter) that controls how strong the cleaning is.

Turn it too low: The noise stays.
Turn it too high: You erase the building details and end up with a smooth, featureless blob.

The authors created two new ways to find the perfect setting for this dial:

The "Teacher" Method (Supervised): If you have a clean version of the map to compare against, the system learns the perfect dial settings by trying to match the teacher's example.
The "Self-Teaching" Method (Unsupervised): If you don't have a clean version, the system uses a trick called Noise2Noise. It takes two different noisy versions of the same map and teaches the system to clean one by looking at the other. It's like trying to fix a blurry photo by comparing it to another blurry photo of the same scene; the system learns to ignore the random static that appears in different places in both photos.

4. Why It's Better (The Graph Filter View)

The authors looked at this mathematically and found that their method acts like a specialized filter.

Old methods were like a sieve that let everything through but smoothed out the rough edges too much (oversmoothing).
Their RED method is like a smart sieve that removes the "static noise" (high-frequency noise) but is careful not to throw away the "sharp details" (important high-frequency features) that make the map look real.

5. The Results

They tested this on two things:

Fake Data: They generated random city maps with noise. Their method cleaned them up better than the old rulebook methods and the standard AI methods.
Real Data: They used 3D models of real objects (like chairs and airplanes). Again, their method removed the noise while keeping the sharp edges of the objects intact, whereas other methods made the objects look blurry or "mushy."

In short: They built a flexible, smart cleaning system for complex data networks. It can use different types of "cleaning experts," it knows how to tune itself perfectly (even without perfect examples), and it keeps the important details sharp while removing the noise.

Technical Summary: Graph Signal Denoising Using Regularization by Denoising and Its Parameter Estimation

Problem Statement
Graph signal processing (GSP) is essential for analyzing data defined on networks, such as 3-D point clouds, sensor networks, and EEG data. However, graph signals are frequently corrupted by noise during measurement. Existing denoising techniques generally fall into two categories: model-based methods (e.g., graph Laplacian regularization) which are interpretable but require manual objective function design, and data-driven methods (e.g., Graph Neural Networks) which perform well with sufficient training data but often lack interpretability and require large datasets. While hybrid approaches like Plug-and-Play Alternating Direction Method of Multipliers (PnP-ADMM) have been proposed to bridge this gap, they often suffer from implicit objective functions that reduce interpretability and require many iterations to converge.

Methodology
This paper proposes a denoising framework for graph signals based on Regularization by Denoising (RED), a technique originally developed for image restoration. The core optimization problem is formulated as:
$\hat{x} = \arg\min_{\tilde{x}} \frac{1}{2}\|\tilde{x} - y\|_2^2 + \frac{\alpha_{red}}{2} \tilde{x}^\top (\tilde{x} - D_{graph}(\tilde{x}))$
where $y$ is the noisy observation, $D_{graph}(\cdot)$ is a graph signal denoiser (acting as a black-box regularizer), and $\alpha_{red}$ is a regularization parameter.

The authors address three primary technical challenges:

Applicability of RED to Graph Signals: The paper theoretically and empirically validates that widely used graph denoisers satisfy the mild conditions required for RED (local homogeneity and strong passivity). Specifically, it proves that Graph Laplacian Regularization (LR) satisfies these conditions under specific graph construction assumptions (e.g., edge weights based on signal distance or Gaussian Markov Random Fields). Experimental results further confirm that Graph Attention Networks (GAT) and PnP-ADMM also satisfy these conditions practically.
Parameter Estimation: To optimize the regularization parameter $\alpha_{red}$ $α_{r e d}$ and the internal denoiser's hyperparameters, the authors propose two training strategies based on Deep Algorithm Unrolling (DAU):
- Supervised: Parameters are learned using ground-truth clean signals via backpropagation through the unrolled iterations.
- Unsupervised: The Noise2Noise framework is adapted to train parameters without clean ground truth by minimizing the error between two noisy versions of the same signal.
Graph Filter Perspective: The authors analyze RED from a spectral perspective, demonstrating that the RED gradient acts as a frequency-dependent scaled version of the internal denoiser's filter. Unlike standard Laplacian regularization which can lead to oversmoothing, RED is shown to preserve high-frequency components while eliminating noise.

Key Contributions

Extension of RED to GSP: This is the first attempt to apply RED to graph signal processing beyond image processing. The paper establishes that many existing graph denoisers, including GNNs, are theoretically or practically compatible with RED.
Interpretable Hybrid Framework: By explicitly incorporating the denoiser into the objective function (rather than just the algorithm steps as in PnP-ADMM), the proposed method maintains interpretability while leveraging the power of data-driven denoisers.
Unsupervised Learning for Graph Signals: The paper introduces an unsupervised parameter estimation method for graph denoising using Noise2Noise, addressing the scarcity of clean ground-truth data in many real-world scenarios.
Algorithmic Implementation: An iterative algorithm (Algorithm 1) is provided to solve the RED optimization problem, with provisions for using conjugate gradient or ADMM.

Experimental Results
The proposed methods were evaluated on synthetic bandlimited graph signals and real-world 3-D point cloud data (ModelNet10).

Performance: In supervised settings, RED-based methods (both with LR and PnP internal denoisers) consistently achieved lower Mean Squared Error (MSE) than standard model-based (LR, PnP) and data-driven (GAT, Untrained GNN) baselines.
DAU Effectiveness: Methods incorporating Deep Algorithm Unrolling (RED-DAU) significantly outperformed their non-unrolled counterparts, demonstrating the benefit of learning iteration-specific parameters.
Unsupervised Performance: The unsupervised RED method (RED-LR-Unsupervised) outperformed existing unsupervised GNN-based methods (GAT, UGNN) despite having significantly fewer parameters.
Visualization: Visual results on 3-D point clouds indicated that RED-based methods effectively reduced noise while preserving sharp features, avoiding the oversmoothing artifacts common in baseline methods.
Computational Cost: The proposed methods exhibited higher computational times compared to simple baselines due to the nested iterative nature of RED and the internal denoiser calls. However, the authors argue this cost is justified by the substantial improvements in denoising accuracy.

Significance and Claims
The paper claims to provide a new, interpretable paradigm for graph signal denoising that effectively balances model-based and data-driven approaches. By demonstrating that RED can be successfully adapted to graph signals and that its parameters can be learned in both supervised and unsupervised settings, the work suggests a path toward more robust and adaptable signal restoration techniques. The authors conclude that their method overcomes existing model-based, data-driven, and hybrid approaches in terms of denoising accuracy, highlighting its potential for future applications in fields requiring high-fidelity signal restoration on complex networks.