Denoising Diffusion Implicit Models (DDIMs)

• DDIMs are non-Markovian generative models that generalize DDPMs by using a deterministic or semi-deterministic reverse process for accelerated sampling.
• The method employs ODE-based and hybrid ODE/SDE sampling strategies to balance generation speed and output diversity.
• Empirical benchmarks and advanced variants demonstrate up to 50× speedup with minimal degradation in sample quality compared to traditional diffusion models.

Denoising Diffusion Implicit Models (DDIMs) are a family of non-Markovian generative models that generalize denoising diffusion probabilistic models (DDPMs) by introducing a deterministic or semi-deterministic reverse process, enabling dramatically accelerated sampling while maintaining high sample quality. DDIMs retain the training objective and forward stochastic process of DDPMs, but decouple the forward and reverse stochasticity at inference, allowing the use of ODE-based or hybrid ODE/SDE sampling. This design results in a tradeoff between sample diversity and generation speed, and underpins several innovations in generative modeling, accelerated diffusion algorithms, and downstream applications.

1. Mathematical Framework: Forward and Reverse Processes

The fundamental structure of DDIMs mirrors that of DDPMs in the forward (noising) process but diverges in the construction and interpretation of the reverse (denoising) mechanism. The forward process is a fixed Markovian chain, adding progressively increasing Gaussian noise to a data sample $x_0$ over $T$ steps:

$$q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{\alpha_t} x_{t-1}, \beta_t I),\quad \bar\alpha_t = \prod_{s=1}^t \alpha_s$$

$$q(x_t | x_0) = \mathcal{N}(x_t; \sqrt{\bar\alpha_t} x_0, (1 - \bar\alpha_t) I)$$

(Song et al., 2020, Berthelot et al., 2023, Wolleb et al., 2022, Shah et al., 2024)

The core departure of DDIMs is in the reverse process. DDIMs define a non-Markovian chain, in which $x_{t-1}$ is an explicit (often deterministic) function of $x_t$ and the network-predicted noise $\epsilon_\theta(x_t, t)$. The canonical deterministic step is:

$$x_{t-1} = \sqrt{\bar\alpha_{t-1}} \hat{x}_0 + \sqrt{1-\bar\alpha_{t-1}} \epsilon_\theta(x_t, t)\,, \quad \text{where } \hat{x}_0 = \frac{x_t - \sqrt{1-\bar\alpha_t} \epsilon_\theta(x_t, t)}{\sqrt{\bar\alpha_t}}$$

A more general form interpolates between deterministic and stochastic behaviors by adding a learned or fixed $\sigma_t z$ noise term:

$$x_{t-1} = \sqrt{\bar\alpha_{t-1}} \hat{x}_0 + \sqrt{1-\bar\alpha_{t-1}-\sigma_t^2} \epsilon_\theta(x_t, t) + \sigma_t z\,,\quad z \sim \mathcal{N}(0, I)$$

Setting $\sigma_t = 0$ yields the fully deterministic (implicit) DDIM chain, corresponding to an ODE-style reverse process (Song et al., 2020, Wolleb et al., 2022, Comanducci et al., 2023).

This formulation allows step-skipping, arbitrary scheduling, and enables non-Markovian trajectories in sampling. The deterministic variant produces a unique output given the same $x_T$, facilitating latent-space interpolation and certain forms of inversion.

2. Theoretical Properties and Algorithmic Variants

The inference dynamics in DDIMs can be interpreted as discretizations of the so-called probability flow ODE associated with the original diffusive SDE:

$$dx = [(\sqrt{1-\beta(t)}-1)x(t) - \tfrac{1}{2}\beta(t) \epsilon_\theta(x(t), t)] \, dt$$

(Liu et al., 2022, Permenter et al., 2023, Zhang et al., 2022)

In this perspective, the deterministic DDIM sampler corresponds to a first-order integration of the probability flow ODE; stochastic variants interpolate between ODE and SDE sampling, with $\eta$ controlling the randomness. Higher-order integration schemes (e.g., the pseudo linear multi-step method, PNDM) have been proposed to further reduce discretization error, leveraging multi-step noise-prediction extrapolations for second-order convergence (Liu et al., 2022, Permenter et al., 2023).

Deterministic DDIMs provide:

• Consistent mapping: $x_T \to x_0$ is bijective (in the noiseless case), enabling semantic interpolation in the latent space.
• Arbitrary step schedules: Non-uniform or truncated inference schedules are possible, with negligible loss in sample quality down to tens of steps (versus 1,000 in DDPM).
• Unified training: Both DDPM and DDIM are trained identically, using the noise-prediction loss: $$L(\theta) = \sum_{t=1}^T \mathbb{E}_{x_0,\epsilon} \left\|\epsilon - \epsilon_\theta(\sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t} \epsilon, t)\right\|_2^2$$ (Song et al., 2020, Shah et al., 2024, Wolleb et al., 2022)
• Fast sampling: Empirically, DDIM yields 10–50× acceleration in wall-clock sampling time with FID degradation of less than 0.5 on CIFAR-10 and CelebA for comparable step counts (Song et al., 2020, Shah et al., 2024).

3. Extensions, Accelerations, and Advanced Scheduling

Several advances build upon the DDIM base:

• gDDIM: Generalizes DDIM to arbitrary linear diffusions, such as non-isotropic SDEs (Blurring Diffusion, Critically-damped Langevin Diffusion), yielding order-of-magnitude acceleration even in these generalized domains (Zhang et al., 2022).
• ShortDF (Shortest-Path Diffusion): Reframes DDIM sampling as a shortest-path optimization in a time-step graph, explicitly optimizing both initial and intermediate residuals to permit aggressive step-skipping and shortcutting. ShortDF achieves comparable or superior sample quality with as few as 2–5 steps (Chen et al., 5 Mar 2025).
• TRACT and BTD: Distillation methods directly compress a long DDIM chain into 1–2 super-steps, reducing compounding teacher errors and supporting strong EMA/SWA generalization. TRACT achieves state-of-the-art single-step FID on CIFAR-10 (3.8 with EDM teacher) and ImageNet64 (7.4), outperforming BTD and vanilla DDIM in the same regime (Berthelot et al., 2023).
• Sawtooth Sampling: For time series, periodic resets of the DDIM schedule further reduce residual error, yielding up to 30× acceleration without reintroducing noise or requiring retraining (Oppel et al., 26 Nov 2025).
• PNDM (Pseudo numerical methods): Treats DDIM as a first-order method on the data manifold and introduces higher-order multi-step variants (PLMS) that reach the same FID as 1000-step DDIMs in only 50 steps (Liu et al., 2022).

These approaches address both theoretical and practical limitations of vanilla diffusion acceleration, offering aggressive sampling with minimal loss or, with proper scheduling, even gains in sample fidelity on challenging distributions.

4. Applications and Conditioning: Image, Audio, Medial, and Inverse Problems

DDIMs provide a foundation for a diverse array of advanced data-generation and manipulation tasks:

• Image generation and translation: Directly yielding high-fidelity generative samples for CIFAR-10, CelebA, ImageNet, and custom image domains (Song et al., 2020, Shah et al., 2024, Zhang et al., 2023, Zhang, 2024).
• Weakly supervised anomaly detection: DDIMs with classifier guidance perform pixel-accurate, class-conditional image-to-image translation, achieving AUROC ≈ 0.90 on BRATS2020 for brain tumor localization and Dice ≃ 0.64 for segmentation, outperforming GAN/VAE baselines (Wolleb et al., 2022).
• Medical image inpainting and augmentation: Modified DDIM architectures conduct bi-directional MS lesion filling and synthesis, supporting realistic lesion-free MR reconstructions and in-silico lesion augmentation for training data (Zhang et al., 2024). CoPaint extends DDIM to Bayesian image inpainting, enforcing pixel constraints via test-time surrogate posterior correction for challenging mask types (Zhang et al., 2023).
• Timbre and audio domain translation: DDIMs are naturally adapted to conditional spectrogram translation (e.g., musical timbre transfer) via log-mel image embedding and deterministic, step-skipped sampling, maintaining musical structure across domains (Comanducci et al., 2023).
• Fully-spiking neuromorphic inference: The FSDDIM method ports the DDIM process into spiking neural nets, using synaptic current learning to operate the chain for energy-efficient sampling on neuromorphic hardware while preserving sample quality (Watanabe et al., 2023).
• Bridge-type generation and structured design: DDIM latent space traversals enable semantic and structural interpolation, producing novel instances (e.g., asymmetric bridges) not present in the training set, by decoding deterministic noise samples through the reverse chain (Zhang, 2024).

Classifier-free guidance (CFG) is directly implemented within the DDIM framework for controllable synthesis, applying simply as a noise-prediction interpolation at each step without architectural changes (Shah et al., 2024). In latent diffusion (VAE + DDIM), the entire chain operates in a compressed representation before decoding, further accelerating the process.

5. Empirical Benchmarks and Quality–Efficiency Tradeoffs

Quantitative analysis across high-profile datasets demonstrates the efficiency and strong generative performance of DDIM and its variants.

Method	Steps	CIFAR-10 FID	CelebA FID	ImageNet64 FID	Speedup Factor
DDPM	1000	≈4.0	≈3.4	–	1×
DDIM	50	≈6.99	≈8.95	–	20×
PLMS (PNDM)	50	≈3.95	≈3.34	–	≥20×
TRACT (EDM T)	1	3.78	–	7.52	1000×
ShortDF	2-10	3.75–9.08	4.30–18.08	–	5–10×

(Song et al., 2020, Wolleb et al., 2022, Berthelot et al., 2023, Liu et al., 2022, Shah et al., 2024, Chen et al., 5 Mar 2025)

Empirically, DDIM-based methods preserve most of the sample fidelity of DDPM while reducing inference time up to 50×. State-of-the-art variants (PLMS, TRACT, ShortDF) can further cut required steps by an order of magnitude or more while matching or surpassing baseline FID.

6. Implementation, Scheduling, and Practical Recommendations

Key implementation best practices for DDIM sampling include:

• Noise schedules: Nonlinear (cosine) schedules for $\beta_t$ smooth training and sample transitions, supporting larger inference steps with minimal artifacts (Shah et al., 2024, Song et al., 2020).
• Inference acceleration: Subsampling the time grid (e.g., $S=20$–$100$) and employing deterministic ODE integration with precomputed $\bar\alpha$ values on-device is optimal.
• Guidance integration: Batch CFG passes with unconditional and conditional networks; replace $\epsilon_\theta$ with the guided interpolation at every step (Shah et al., 2024).
• Latent domain operation: For large images or constrained memory, apply DDIM to VAE latents and decode after denoising. Latent noise-prediction loss must be adjusted for the compression dimension (Shah et al., 2024).
• Distillation: For ultra-fast or single-step generation, distilled architectures (TRACT, BTD) compress multi-step chains into minimal inference steps; momentum selection for EMA/SWA is crucial for robust training (Berthelot et al., 2023).

Recommendations: Use cosine schedules for improved stability, refine batch size and memory settings for large-scale inference, and employ knowledge distillation or multi-step methods for applications requiring real-time sampling.

7. Limitations, Extensions, and Ongoing Research

Despite their advantages, DDIMs entail trade-offs:

• Deterministic trajectory limitations: Fully deterministic samplers can limit output diversity and poorly explore multimodal posteriors.
• Underfitting with aggressive step-skipping: Excessive reduction in step count (e.g., 1-step sampling) without careful design or distillation can significantly degrade sample quality, as seen in 1-phase TRACT ablations (FID ≈14.4) (Berthelot et al., 2023).
• Data manifold assumptions: DDIM’s stability is underpinned by locality and smoothness of the denoiser’s score on data-like manifolds; large jumps across low-density regions may fail.
• Inversion and editing: The deterministic mapping supports interpolation but does not guarantee invertibility for arbitrary images or full editability; embedding strategies remain an open research area.

Active research directions include higher-order integration (PNDM, second-order gradient estimators (Liu et al., 2022, Permenter et al., 2023)), transitive-closure and graph-theoretical acceleration (TRACT, ShortDF), and application-specific adaptations (neuromorphic, time-series, anomaly detection).

In summary, DDIMs represent a significant advance in generative diffusion modeling, providing a versatile, efficient, and theoretically principled alternative to Markovian sampling, with extensive adaptation and benchmarking across data domains and methodological frontiers.