Pseudo Numerical Methods for Diffusion Models on Manifolds (2202.09778v2)

Abstract: Denoising Diffusion Probabilistic Models (DDPMs) can generate high-quality samples such as image and audio samples. However, DDPMs require hundreds to thousands of iterations to produce final samples. Several prior works have successfully accelerated DDPMs through adjusting the variance schedule (e.g., Improved Denoising Diffusion Probabilistic Models) or the denoising equation (e.g., Denoising Diffusion Implicit Models (DDIMs)). However, these acceleration methods cannot maintain the quality of samples and even introduce new noise at a high speedup rate, which limit their practicability. To accelerate the inference process while keeping the sample quality, we provide a fresh perspective that DDPMs should be treated as solving differential equations on manifolds. Under such a perspective, we propose pseudo numerical methods for diffusion models (PNDMs). Specifically, we figure out how to solve differential equations on manifolds and show that DDIMs are simple cases of pseudo numerical methods. We change several classical numerical methods to corresponding pseudo numerical methods and find that the pseudo linear multi-step method is the best in most situations. According to our experiments, by directly using pre-trained models on Cifar10, CelebA and LSUN, PNDMs can generate higher quality synthetic images with only 50 steps compared with 1000-step DDIMs (20x speedup), significantly outperform DDIMs with 250 steps (by around 0.4 in FID) and have good generalization on different variance schedules. Our implementation is available at https://github.com/luping-liu/PNDM.

Pseudo Numerical Methods for Diffusion Models on Manifolds

The paper "Pseudo Numerical Methods for Diffusion Models on Manifolds" by Luping Liu et al. investigates a novel approach to enhance the efficiency and effectiveness of Denoising Diffusion Probabilistic Models (DDPMs). These models are renowned for generating high-quality samples such as images and audio; however, they traditionally require a large number of iterations for sampling, which limits practical applicability.

Overview

The authors propose treating DDPMs as a process of solving differential equations on manifolds and introduce what they term "pseudo numerical methods" (PNDMs). This fresh perspective allows for reformulating the inference process, leading to both a reduction in computational cost and an improvement in sample quality. The paper highlights that previous acceleration methods often compromise on the quality due to new noise introduction.

Key Contributions

• PNDMs as a Generalization: The methodology redefines diffusion models in terms of solving differential equations on manifolds. This approach extends existing methods like Denoising Diffusion Implicit Models (DDIMs) and incorporates classical numerical methods adapted to manifolds.
• Experiments and Results: The proposed method achieves a remarkable performance improvement on datasets such as Cifar10, CelebA, and LSUN. Specifically, PNDMs generate high-quality images with only 50 steps, outperforming traditional DDIMs by a significant margin in FID scores—a key performance metric in image generation.
• Numerical Analysis: The paper presents detailed theoretical analyses of pseudo numerical methods, confirming second-order convergence. This underscores the method's efficiency and accuracy.

Implications and Future Directions

The implications of this research are manifold. Practically, it paves the way for faster generation of synthetic data, reducing computational time without sacrificing quality. Theoretically, it enriches the understanding of diffusion processes in generative modeling by framing them within the context of manifold theory.

Future developments could explore optimizing the variance schedules specific to PNDMs, potentially enhancing their performance. Moreover, extending PNDMs to wider applications beyond image generation, such as in neural ODEs, presents an exciting frontier.

In summary, the paper introduces a nuanced approach to handling DDPMs through pseudo numerical methods, yielding notable gains in speed and quality. The thorough analytical and experimental validation supports its potential utility in advancing the field of generative modeling.