Reverse Diffusion in Score-Based Generative Models

1. The Reverse Problem and the Score Function

The core challenge is to estimate the reverse conditional probability $p({\mathbf{x}}_{t-1}|{\mathbf{x}}_t)$. This distribution is intractable because it would require knowing the distribution of all possible images. However, we can approximate it with a neural network.

Instead of directly modeling this complex distribution, the score-based approach re-frames the problem: what if we could just find a function that tells us how to "nudge" a noisy image ${\mathbf{x}}_t$ to make it slightly more likely under the true data distribution?

This is precisely what the score function does. The score of a probability density function $p(\mathbf{x})$ is defined as the gradient of its log-probability with respect to the data $\mathbf{x}$:

Intuitively, the score function at any point $\mathbf{x}$ gives you a vector pointing in the direction of the steepest ascent on the log-probability landscape—that is, it points toward regions where the data is more dense or "more likely".

If we could learn the score function $s_t({\mathbf{x}}_t)={\nabla }_{{\mathbf{x}}_t}\log p({\mathbf{x}}_t)$ for the distribution of noisy data at every timestep $t$, we would have a "compass" to guide us from pure noise back to a clean image.

The following resource provides a concise introduction to this idea.

2. Learning the Score with Denoising Score Matching

This is great in theory, but we have a problem: we don't know $p_t({\mathbf{x}}_t)$, so we can't compute its score to use as a training target. The first attempt to solve this is a technique called Score Matching, which minimizes the squared distance between a model's score $s_{\theta }(\mathbf{x})$ and the true score ${\nabla }_{\mathbf{x}}\log p_{data}(\mathbf{x})$. However, this objective is itself intractable to compute directly.

The breakthrough came with Denoising Score Matching. Instead of matching the score of the clean data, we match the score of a noise-perturbed version of the data. This connects directly to the diffusion process.

Let's say we perturb a clean image ${\mathbf{x}}_0$ by adding Gaussian noise $\mathbf{ϵ}$ to get ${\mathbf{x}}_t$, just like in our forward process. The conditional distribution $q({\mathbf{x}}_t|{\mathbf{x}}_0)$ is a simple Gaussian. It turns out that the score of this conditional distribution, ${\nabla }_{{\mathbf{x}}_t}\log q({\mathbf{x}}_t|{\mathbf{x}}_0)$, has a simple, closed-form solution:

This is a remarkable result. The score—the direction towards the original clean image ${\mathbf{x}}_0$ from the noisy image ${\mathbf{x}}_t$—is directly proportional to the negative of the noise that was added. This makes perfect intuitive sense: to denoise the image, we must "subtract" the noise that was added.

Our training objective now becomes tractable. We train a neural network, $s_{\theta }({\mathbf{x}}_t,t)$, to predict this score. The loss function is essentially a regression task:

If we reparameterize our network to predict the noise $\mathbf{ϵ}$ instead of the score (since they are proportional), we arrive at an objective that is almost identical to the simplified training loss used in DDPMs:

Here, ${\mathbf{ϵ}}_{\theta }$ is a neural network (typically a U-Net) that takes the noisy image ${\mathbf{x}}_t$ and the timestep $t$ as input and tries to predict the noise $\mathbf{ϵ}$ that was used to create it. This is exactly the objective you would have implemented in your basic DDPM. Now you see its connection to the more fundamental concept of score matching.

To see the key steps of this derivation and how it connects score matching to denoising, the following video provides a very clear explanation.

3. A Continuous Perspective: Stochastic Differential Equations (SDEs)

The score-based formulation allows us to generalize the discrete diffusion process into a continuous-time framework using Stochastic Differential Equations (SDEs). This provides a more powerful and flexible theoretical foundation.

Think of the forward process not as $T$ discrete steps, but as a continuous evolution from time $t=0$ to $t=T$. This process can be described by a forward SDE:

• $\mathbf{f}(\mathbf{x},t)$ is the drift coefficient, a deterministic term that pulls the data towards a certain point (e.g., the origin).
• $g(t)$ is the diffusion coefficient, which controls the magnitude of the random noise $d\mathbf{w}$ being added at time $t$.

The amazing insight, first established by Anderson in 1982, is that any such forward SDE has a corresponding reverse-time SDE that perfectly reverses the process. This reverse SDE is given by:

Look closely at the drift term. To run the reverse process, we need the original drift, the diffusion coefficient, and our old friend: the score function ${\nabla }_{{\mathbf{x}}_t}\log p_t({\mathbf{x}}_t)$. This is the key that unlocks the reverse journey from noise to data in this continuous framework. Since we've already trained a network ${\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)$ that can approximate this score, we can now numerically solve this reverse SDE to generate new samples.

The following resources will help you solidify this concept.

For a more formal and textual explanation, the following article is very direct and clear.

4. Unifying DDPM and Score-Based Models

So, how does this all relate back to the DDPM you've implemented?

• The discrete forward process of a DDPM is a discretization of a specific type of forward SDE.
• The DDPM's reverse process, which involves a neural network predicting the noise ${\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t)$, is equivalent to learning the score function.
• The standard DDPM sampler (ancestral sampling) is one of the simplest possible numerical solvers (Euler-Maruyama) for the corresponding reverse-time SDE.

The SDE framework reveals that DDPM is just one specific choice of SDE and solver. This perspective is incredibly powerful because it means we can:

• Design different forward SDEs (e.g., Variance Exploding vs. Variance Preserving).
• Use more advanced SDE solvers (like DDIM or DPM-Solver, which we'll cover in the next module) to generate samples much faster and more accurately.

The following video provides a fantastic summary that compares the two viewpoints side-by-side.

Conclusion

This lesson was more theoretical than the last, but it has equipped you with a much deeper understanding of how and why diffusion models work.

Here are the key takeaways:

• The reverse diffusion process can be framed as learning a score function, ${\nabla }_{\mathbf{x}}\log p(\mathbf{x})$, which provides a vector field guiding samples from low-density to high-density regions.
• Denoising Score Matching is a tractable method for learning this score, and its objective is equivalent to training a network to predict the noise added during the forward process—the same objective used in DDPMs.
• The entire diffusion process can be generalized to a continuous-time framework using Stochastic Differential Equations (SDEs).
• A reverse-time SDE, which depends critically on the score function, defines the generative process, transforming noise back into data.
• This SDE framework unifies DDPMs and score-based models, showing that a DDPM is one specific discretization of a more general process. This view opens the door to many advanced sampling and modeling techniques.

In our next lesson, we will return to the original DDPM paper's perspective and analyze the variational lower bound (ELBO). This will complete our understanding of the theoretical underpinnings of the DDPM loss function from the variational inference point of view, before we move on to exploring more advanced architectures and samplers.