Trade-offs Between DDPM and DDIM

You've learned about two primary methods for sampling from a trained diffusion model: Denoising Diffusion Probabilistic Models (DDPM) and Denoising Diffusion Implicit Models (DDIM). While both leverage the trained noise prediction network to reverse the diffusion process, they operate differently, leading to distinct trade-offs. Choosing between them depends on your specific needs regarding generation speed, sample quality, and the desired level of randomness.

Let's compare DDPM and DDIM across several important dimensions.

Generation Speed

DDPM sampling, as originally formulated, requires simulating the reverse Markov chain step-by-step. If the model was trained for $T$ timesteps (often $T=1000$ or more), DDPM sampling involves $T$ sequential evaluations of the neural network. This makes the process computationally intensive and relatively slow, as each step depends on the previous one.

DDIM offers a significant advantage here. Because DDIM formulates the generation process differently (related to solving an underlying differential equation), it doesn't strictly require simulating every single timestep from the original forward process sequence $1, ..., T$. Instead, you can choose a subsequence of timesteps, say $S < T$ steps (e.g., $S=50, 100, 200$), and perform the denoising updates only at these selected timesteps. This drastically reduces the number of required network evaluations, leading to much faster sample generation. For instance, using DDIM with 100 steps can be 10 times faster than using DDPM with 1000 steps.

Sample Quality

Generally, DDPM sampling with a large number of steps ($T=1000$ or more) is known to produce high-fidelity samples that closely match the training data distribution. The slow, gradual denoising process with added stochasticity at each step contributes to this quality.

DDIM can often achieve comparable, or at least very good, sample quality with significantly fewer steps than DDPM. However, there's usually a trade-off: reducing the number of DDIM steps too much (e.g., below 20-50, depending on the model and dataset) can lead to a noticeable degradation in sample quality compared to a full DDPM run. The optimal number of DDIM steps often needs experimentation to balance speed and fidelity. For many applications, the quality from DDIM with 100-200 steps is sufficient and the speed gain is substantial.

Hypothetical comparison showing how sample quality (represented by a metric like FID where lower is better) might change with the number of sampling steps for DDPM and DDIM. DDIM achieves good quality much faster but might have a slightly higher floor than DDPM with maximum steps.

Determinism vs. Stochasticity

DDPM sampling is inherently stochastic. At each step $t$, the reverse transition $p_\theta(x_{t-1}|x_t)$ involves sampling, typically by adding Gaussian noise scaled by a variance term $\sigma_t^2$. This means that even starting from the same initial noise $x_T$, running the DDPM sampling process multiple times will produce different final samples $x_0$. This stochasticity contributes to the diversity of generated samples.

DDIM introduces a parameter, often denoted as $\eta$ (eta), which controls the amount of stochasticity in the sampling process.

• When $\eta = 1$, the DDIM process behaves similarly to DDPM in terms of stochasticity (though using the DDIM update rule).
• When $\eta = 0$, the DDIM sampling process becomes deterministic. Given the same initial noise $x_T$, running DDIM with $\eta=0$ will always produce the exact same final sample $x_0$. This is because the random noise term in the update step vanishes when $\eta=0$.

This deterministic property ($\eta=0$) is useful for applications requiring reproducibility or for tasks like image inversion and manipulation where you want a predictable mapping between the latent noise and the generated image. Values between 0 and 1 allow interpolating between deterministic and stochastic generation.

Flexibility in Step Sizes

DDPM sampling follows the predefined sequence of timesteps $T, T-1, ..., 1$. It doesn't easily allow for varying the "size" of the steps taken during denoising.

DDIM's formulation provides more flexibility. By allowing sampling using a subsequence of the original timesteps, DDIM effectively allows for larger, non-uniform steps in the denoising process. This is mathematically justified by its connection to solving differential equations, where step sizes can be adapted.

Summary Table

Feature	DDPM	DDIM
Speed	Slow (Requires T steps)	Fast (Can use S << T steps)
Quality	High (especially with many steps)	Good (Often comparable with fewer steps, degrades if too few)
Stochasticity	Always Stochastic	Controlled by $\eta$; Deterministic if $\eta=0$
Step Size	Fixed (Follows original T steps)	Flexible (Uses subsequence of steps)
Use Case	Max fidelity, diversity generation	Faster generation, interactive use, deterministic outputs

Choosing Between DDPM and DDIM:

• If your primary goal is to generate the highest possible quality samples and computational time is not a major constraint, DDPM (with a large T) is a strong choice.
• If you need faster generation, for example, for interactive applications or generating large batches of images quickly, DDIM (with a moderate number of steps, e.g., 50-200) is generally preferred.
• If you require deterministic outputs from the same starting noise (e.g., for reproducibility or controlled editing), DDIM with $\eta=0$ is the method to use.

In practice, DDIM is very widely used due to its significant speed advantage while often maintaining excellent sample quality. Understanding these trade-offs allows you to select the sampling method that best aligns with the requirements of your specific generative task.