💻 computer science

DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions

This paper proposes DEGMC, a novel denoising diffusion model that enhances geometric feature extraction and network equivariance by incorporating Riemannian group morphological convolutions derived from Hamilton-Jacobi partial differential equations.

Original authors: El Hadji S. Diop, Thierno Fall, Mohamed Daoudi

Published 2026-02-12

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CC BY 4.0

Original authors: El Hadji S. Diop, Thierno Fall, Mohamed Daoudi

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 are trying to teach a robot how to draw a perfect sketch of a cat.

Currently, most AI models (like the ones used to generate images) learn by a process called "Diffusion." Think of this like taking a beautiful drawing and slowly sprinkling sand over it until the drawing is completely buried and you can’t see anything but a pile of sand. The AI’s job is to learn how to "un-sand" the image—to look at a pile of sand and figure out exactly how to brush it away to reveal the cat underneath.

However, this paper identifies two big problems with how current "un-sanding" robots work.

The Two Problems

The "Orientation" Problem (Lack of Equivariance):
Imagine if you taught your robot to recognize a cat only when it was sitting upright. If the cat is lying on its side or upside down, the robot gets confused and thinks it’s looking at a pile of sand instead of a cat. Most current AI models are "translation equivariant," which is a fancy way of saying they are good at recognizing things if they move left, right, up, or down, but they struggle if the object rotates or flips.
The "Fine Detail" Problem (Geometric Extraction):
When the AI is brushing away the sand, it often struggles with the tiny, delicate parts—like a single whisker or the sharp edge of an ear. It tends to "smudge" the fine details because it doesn't truly understand the geometry (the shape and structure) of what it's looking at.

The Solution: DEGMC (The "Smart Brush" Approach)

The authors of this paper created a new method called DEGMC. Instead of giving the robot a standard, blunt brush, they gave it a "Smart Geometric Brush" that understands shapes and symmetries.

Here is how their "Smart Brush" works, using three special tools:

1. The "Convection" Tool (The Navigator)

Instead of just brushing randomly, this tool helps the AI understand where things are moving. It acts like a GPS for the brush, allowing it to follow the natural curves and flows of the image. If a whisker is curved, the brush follows that curve rather than trying to brush straight through it.

2. The "Dilation and Erosion" Tools (The Sculptors)

In the world of math, "dilation" is like adding a little bit of clay to a shape to make it fuller, and "erosion" is like carving away a little bit to make it sharper.

The Analogy: Imagine you are sculpting a statue out of sand. The AI uses these tools to "expand" the important features (like the eyes) and "sharpen" the edges (like the ears). Because these tools are "Equivariant," the robot knows that a "sharpening" motion works exactly the same way whether the cat is upright, sideways, or upside down.

3. The "Hyperbolic" Playground (The Curved Map)

The researchers didn't just teach the AI on a flat table. They taught it on a "Hyperbolic Manifold"—which you can imagine as a curved, infinite saddle-shaped surface.

The Analogy: On a flat map, distances can get distorted. But on this special curved surface, the AI can "fold" complex information more efficiently. It’s like giving the AI a 3D brain to solve a 2D problem, allowing it to see the relationships between parts of an image much more clearly.

The Result: A Better Artist

When the researchers tested this "Smart Brush" (the DEGMC model) against the old way (the standard DDPM model), they found three big wins:

It’s Faster: The AI doesn't need as much practice to get good. It "gets it" much earlier in its training.
It’s Smarter with Rotations: When they took the test images and rotated them (the "RotoMNIST" test), the old AI got confused and produced blurry messes. The new DEGMC model didn't care—it saw the cat just as clearly, no matter the angle.
It’s Sharper: The images produced (even complex color ones like CIFAR-10) had better details and more variety.

In short: The paper moves AI from being a "blind brush" that just wipes away noise, to a "master sculptor" that understands the geometry, symmetry, and delicate structure of the world.

Technical Summary: DEGMC (Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions)

1. Problem Statement

The paper addresses two fundamental limitations in current Denoising Diffusion Probabilistic Models (DDPM), specifically those utilizing the standard U-Net architecture:

Limited Equivariance: Standard U-Nets are theoretically only translation-equivariant. They fail to intrinsically account for other geometric symmetries such as rotations, reflections, and permutations (the Euclidean group $E(n)$ ).
Geometric Feature Extraction: Traditional convolutional layers often struggle to capture fine, non-linear geometric structures and thin features, which are critical for high-fidelity image and manifold-based data generation.

2. Methodology

The authors propose DEGMC, a framework that integrates Riemannian geometry and group theory into the diffusion reverse process. The core innovation lies in replacing standard convolutional layers with equivariant group morphological convolutions and convection operators.

Key Technical Components:

Riemannian Manifold Framework: The model is defined on general Riemannian manifolds. For computational implementation, the authors utilize the Hyperbolic Ball ( $\mathbb{B}^n$ ). This choice is strategic because the hyperbolic metric is invariant under the Euclidean group $E(n)$ , allowing the model to preserve symmetries like rotation and reflection while capturing hierarchical data structures through negative curvature.
Convection-Dilation-Erosion (CDE) Blocks: Instead of standard ResNet blocks, the authors introduce ResnetCDEBlocks in the U-Net's middle block. These blocks consist of:
- Convection Term: Solved via the method of characteristics using a $G$ -invariant vector field. This acts as a learned resampling operator to align features spatially.
- Morphological Dilations and Erosions: Derived from the viscosity solutions of first-order Hamilton-Jacobi type PDEs. These operators allow for multiscale geometric processing. Unlike standard convolutions, these morphological operations can better represent non-linearities and thin geometric structures.
GMCUnet Architecture: A modified U-Net where the encoder and decoder handle multiscale features, but the critical "middle block" (the bottleneck) utilizes the CDE blocks to enforce $E(n)$ equivariance.

3. Key Contributions

Novel Architecture: The introduction of the GMCUnet, the first reported combination of a diffusion U-Net with CDE-type residual blocks.
Mathematical Integration: A formal framework that combines the probabilistic nature of DDPM with the geometric rigor of Hamilton-Jacobi PDEs and Lie group actions.
Symmetry Preservation: A systematic approach to ensuring the model is equivariant to translations, rotations, reflections, and permutations, thereby reducing the complexity of the learning task.

4. Experimental Results

The model was evaluated against the baseline DDPM on three datasets: MNIST, RotoMNIST (rotated MNIST), and CIFAR-10.

MNIST: DEGMC showed faster convergence, reaching high-quality sample generation in fewer iterations than the baseline.
RotoMNIST (The Stress Test): This dataset highlights the model's primary strength. While the baseline DDPM's performance dropped significantly due to rotations, DEGMC maintained high quality, achieving a much lower FID (35.75) compared to DDPM (44.74). This proves the effectiveness of the intrinsic equivariance.
CIFAR-10: On real-world color images, DEGMC outperformed the baseline in both FID (lower is better) and Inception Score (IS) (higher is better), indicating superior image fidelity and greater sample diversity.

5. Significance

The significance of this work lies in its shift from purely statistical denoising to geometry-aware denoising. By embedding the physical and geometric symmetries of the data directly into the neural network architecture via PDEs, the authors demonstrate that:

Models can be made robust to geometric transformations without requiring massive data augmentation.
Mathematical operators (morphology/convection) can serve as powerful, interpretable alternatives to standard learnable weights for capturing complex structural features.
The framework provides a scalable path toward applying diffusion models to more complex domains, such as 3D molecular modeling and non-Euclidean manifold data.