Pseudo Numerical Methods for Diffusion Models on Manifolds

TL;DR

This paper introduces pseudo numerical methods for diffusion models, treating them as differential equations on manifolds, which significantly accelerates sample generation while maintaining high quality.

Contribution

It proposes a novel perspective of solving diffusion models as differential equations on manifolds and introduces pseudo numerical methods, improving speed and quality over existing methods.

Findings

PNDM generates higher quality images with 50 steps compared to 1000-step DDIMs.

PNDM outperforms DDIMs with 250 steps by around 0.4 in FID.

PNDM demonstrates good generalization across different variance schedules.

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.