To understand the landscape of modern generative diffusion models, you need to compare three influential frameworks: Denoising Diffusion Probabilistic Models (DDPMs), Score-based SDEs (Score-SDEs), and the broader class of score-based generative models. Each of these approaches leverages the concept of gradually adding noise to data and then learning to reverse this process, but they differ in their mathematical formalism and generative procedures.

DDPMs are discrete-time models that define a forward process as a Markov chain of Gaussian noise additions, and a reverse process parameterized to stepwise denoise. The model learns to predict either the original data or the noise added at each timestep, using a variational lower bound objective.

Score-based generative models generalize this idea by learning the gradient of the data distribution's log-density (the "score") at various noise levels. This score function enables the construction of generative samplers, for example via Langevin dynamics, to iteratively sample from the data distribution.

Score-SDEs further extend this by modeling the forward diffusion as a continuous-time stochastic differential equation (SDE). The reverse process is then also an SDE, parameterized by the learned score function. This continuous view allows for flexible sampling algorithms, including probability flow ODEs and various SDE solvers.

Mathematically, the mapping between these formulations can be made explicit. In DDPMs, the forward process is defined as a sequence of Gaussian transitions:

where $beta_t$ is the variance schedule for timestep $t$. The reverse process is parameterized as:

Score-based generative models instead focus on learning $s_{\theta}(x, \sigma)$, the score function at noise level $\sigma$, by minimizing a score matching loss.

Score-SDEs generalize the diffusion to continuous time:

with $f$ and $g$ defining the drift and diffusion coefficients, and $w$ standard Brownian motion. The reverse SDE is:

where $\nabla_x log (p_t(x))$ is the score function, learned by a neural network.

The discrete-time DDPM can be viewed as a special case of a continuous SDE with piecewise-constant coefficients, while the score-based approach provides a unifying view: both DDPM and Score-SDE learn the score function, but differ in how they use it for sampling — discrete Markov steps versus continuous SDE/ODE integration.

To clarify these relationships, consider a conceptual example. Imagine you want to generate realistic images of handwritten digits.

• In the DDPM approach, you would start from pure Gaussian noise and iteratively denoise the image in discrete steps, each time using a neural network to predict the noise or the clean image;
• In a score-based model using Langevin dynamics, you would also start from noise, but take many small steps in the direction given by the score function to gradually move the sample toward a realistic digit;
• In the Score-SDE framework, you would treat the image generation as solving a continuous-time SDE, where the learned score function guides the reverse-time evolution from noise to data.

All three methods rely on learning how to denoise, but DDPM uses fixed discrete steps, score-based models use the score for iterative refinement, and Score-SDEs use a continuous stochastic process. Understanding these similarities and differences helps you choose the right formulation for your generative modeling task.