Diffusion Bridge Implicit Models

## Abstract

Abstract Denoising diffusion bridge models (DDBMs) are a powerful variant of diffusion models for interpolating between two arbitrary paired distributions given as endpoints. Despite their promising performance in tasks like image translation, DDBMs require a computationally intensive sampling process that involves the simulation of a (stochastic) differential equation through hundreds of network evaluations. In this work, we take the first step in fast sampling of DDBMs without extra training, motivated by the well-established recipes in diffusion models. We generalize DDBMs via a class of non-Markovian diffusion bridges defined on the discretized timesteps concerning sampling, which share the same marginal distributions and training objectives, give rise to generative processes ranging from stochastic to deterministic, and result in diffusion bridge implicit models (DBIMs). DBIMs are not only up to 25 × \times × faster than the vanilla sampler of DDBMs but also induce a novel, simple, and insightful form of ordinary differential equation (ODE) which inspires high-order numerical solvers. Moreover, DBIMs maintain the generation diversity in a distinguished way, by using a booting noise in the initial sampling step, which enables faithful encoding, reconstruction, and semantic interpolation in image translation tasks. Code is available at https://github.com/thu-ml/DiffusionBridge .

## 1 Introduction

Diffusion models (Song et al., ; Sohl-Dickstein et al., ; Ho et al., ) represent a family of powerful generative models, with high-quality generation ability, stable training, and scalability to high dimensions. They have consistently obtained state-of-the-art performance in various domains, including image synthesis (Dhariwal & Nichol, ; Karras et al., ), speech and video generation (Chen et al., ; Ho et al., ), controllable image manipulation (Nichol et al., ; Ramesh et al., ; Rombach et al., ; Meng et al., ), density estimation (Song et al., ; Kingma et al., ; Lu et al., ; Zheng et al., ) and inverse problem solving (Chung et al., ; Kawar et al., ). They also act as fundamental components of modern text-to-image (Rombach et al., ) and text-to-video (Gupta et al., ; Bao et al., ) synthesis systems, ushering in the era of AI-generated content.

However, diffusion models are not well-suited for solving tasks like image translation or restoration, where the transport between two arbitrary probability distributions is to be modeled given paired endpoints. Diffusion models are rooted in a stochastic process that gradually transforms between data and noise, and the prior distribution is typically restricted to the “non-informative” random Gaussian noises. Adapting diffusion models to scenarios where a more informative prior naturally exists, such as image translation/restoration, requires cumbersome techniques like adding conditions to the model (Ho & Salimans, ; Saharia et al., ), modifying the generation pipeline (Meng et al., ; Su et al., ) or adding extra guidance terms during sampling (Chung et al., ; Kawar et al., ). These approaches inevitably result in either sub-par performance or slow and resource-intensive inference processes.

Recently, denoising diffusion bridge models (DDBMs) (Zhou et al., ) have emerged as a scalable and promising approach to solving the distribution translation tasks. By considering the reverse-time processes of a diffusion bridge, which represent diffusion processes conditioned on given endpoints, DDBMs offer a general framework for distribution translation. While excelling in image translation tasks with exceptional quality and fidelity, sampling from DDBMs requires simulating a (stochastic) differential equation corresponding to the reverse-time process. Even with the introduction of their hybrid sampler, achieving high-fidelity results for high-resolution images still demands over 100 steps. Compared to the efficient samplers for diffusion models (Song et al., ; Zhang & Chen, ; Lu et al., ), which require around 10 steps to generate reasonable samples, DDBMs are falling behind, urging the development of efficient variants.

Figure: Figure 1: Inpainting results on the ImageNet 256$\times$256 dataset (Deng et al., ) by DDBM (Zhou et al., ) with 100 number of function evaluations (NFE), and DBIM (ours) with only 10 NFE.
Refer to caption: extracted/5978797/figures/condition.png

This work represents the first pioneering effort toward accelerated sampling of DDBMs. As suggested by well-established recipes in diffusion models, training-free accelerations of diffusion sampling primarily focus on reducing stochasticity (e.g. the prominent denoising diffusion implicit models, DDIMs) and utilizing higher-order information (e.g. high-order solvers). We present diffusion bridge implicit models (DBIMs) as an approach that explores both aspects within the diffusion bridge framework. Firstly, we investigate the continuous-time forward process of DDBMs on discretized timesteps and generalize them to a series of non-Markovian diffusion bridges controlled by a variance parameter, while maintaining identical marginal distributions and training objectives as DDBMs. Secondly, the induced reverse generative processes correspond to sampling procedures of varying levels of stochasticity, including deterministic ones. Consequently, DBIMs can be viewed as a bridge counterpart and extension of DDIMs.
Furthermore, in the continuous time limit, DBIMs can induce a novel form of ordinary differential equation (ODE), which is linked to the probability flow ODE (PF-ODE) in DDBMs while being simpler and significantly more efficient. The induced ODE also facilitates novel high-order numerical diffusion bridge solvers for faster convergence.

We demonstrate the superiority of DBIMs by applying them in image translation and restoration tasks, where they offer up to 25$\times$ faster sampling compared to DDBMs and achieve state-of-the-art performance on challenging high-resolution datasets. Unlike conventional diffusion sampling, the initial step in DBIMs is forced to be stochastic with a booting noise to avoid singularity issues arising from the fixed starting point on a bridge. By viewing the booting noise as the latent variable, DBIMs maintain the generation diversity of typical generative models while enabling faithful encoding, reconstruction, and semantically meaningful interpolation in the data space.

## 2 Background

### 2.1 Diffusion Models

Given a $d$-dimensional data distribution $q_{0}(\bm{x}_{0})$, diffusion models (Song et al., ; Sohl-Dickstein et al., ; Ho et al., ) build a diffusion process by defining a forward stochastic differential equation (SDE) starting from $\bm{x}_{0}\sim q_{0}$:

$$ $\mathrm{d}\bm{x}_{t}=f(t)\bm{x}_{t}\mathrm{d}t+g(t)\mathrm{d}\bm{w}_{t}$ (1) $$

where $t\in[0,T]$ for some finite horizon $T$, $f,g:[0,T]\to\mathbb{R}$ is the scalar-valued drift and diffusion term, and $\bm{w}_{t}\in\mathbb{R}^{d}$ is a standard Wiener process. As a linear SDE, the forward process owns an analytic Gaussian transition kernel

$$ $q_{0t}(\bm{x}_{t}|\bm{x}_{0})=\mathcal{N}(\alpha_{t}\bm{x}_{0},\sigma_{t}^{2} \bm{I})$ (2) $$

by Itô’s
formula (Itô, ), where
$\alpha_{t},\sigma_{t}$ are called noise schedules satisfying $f(t)=\frac{\mathrm{d}\log\alpha_{t}}{\mathrm{d}t},\quad g^{2}(t)=\frac{\mathrm
{d}\sigma_{t}^{2}}{\mathrm{d}t}-2\frac{\mathrm{d}\log\alpha_{t}}{\mathrm{d}t}
\sigma_{t}^{2}$ (Kingma et al., ). The forward SDE is accompanied by a series of marginal distributions $\{q_{t}\}_{t=0}^{T}$ of $\{\bm{x}_{t}\}_{t=0}^{T}$, and $f,g$ are properly designed so that the terminal distribution is approximately a pure Gaussian, i.e., $q_{T}(\boldsymbol{x}_{T})\approx\mathcal{N}(\bm{0},\sigma_{T}^{2}\bm{I})$.

To sample from the data distribution $q_{0}(\bm{x}_{0})$, we can solve the reverse SDE or probability flow ODE (Song et al., ) from $t=T$ to $t=0$:

$$ $\displaystyle\mathrm{d}\bm{x}_{t}$ $\displaystyle=[f(t)\bm{x}_{t}-g^{2}(t)\nabla_{\bm{x}_{t}}\log q_{t}(\bm{x}_{t} )]\mathrm{d}t+g(t)\mathrm{d}\bar{\bm{w}}_{t},$ (3) $\displaystyle\mathrm{d}\bm{x}_{t}$ $\displaystyle=\left[f(t)\bm{x}_{t}-\frac{1}{2}g^{2}(t)\nabla_{\bm{x}_{t}}\log q _{t}(\bm{x}_{t})\right]\mathrm{d}t.$ (4) $$

They share the same marginal distributions $\{q_{t}\}_{t=0}^{T}$ with the forward SDE, where $\bar{\boldsymbol{w}}_{t}$ is the reverse-time Wiener process, and the only unknown term $\nabla_{\bm{x}_{t}}\log q_{t}(\bm{x}_{t})$ is the score function of the marginal density $q_{t}$. By denoising score matching (DSM) (Vincent, ), a score prediction network $\bm{s}_{\theta}(\bm{x}_{t},t)$ can be parameterized to minimize $\mathbb{E}_{t}\mathbb{E}_{\bm{x}_{0}\sim q_{0}(\bm{x}_{0})}\mathbb{E}_{\bm{x}_
{t}\sim q_{0t}(\bm{x}_{t}|\bm{x}_{0})}\left[w(t)\|\bm{s}_{\theta}(\bm{x}_{t},t
)-\nabla_{\bm{x}_{t}}\log q_{0t}(\bm{x}_{t}|\bm{x}_{0})\|_{2}^{2}\right]$, where $q_{0t}$ is the analytic forward transition kernel and $w(t)$ is a positive weighting function. $\bm{s}_{\theta}$ can be plugged into the reverse SDE and the probability flow ODE to obtain the parameterized diffusion SDE and diffusion ODE. There are various dedicated solvers for diffusion SDE or ODE (Song et al., ; Zhang & Chen, ; Lu et al., ; Gonzalez et al., ).

### 2.2 Denoising Diffusion Bridge Models

Denoising diffusion bridge models (DDBMs) (Zhou et al., ) consider driving the diffusion process in Eqn. (1) to arrive at a particular point $\bm{y}\in\mathbb{R}^{d}$ almost surely via Doob’s h-transform (Doob & Doob, ):

$$ $\mathrm{d}\bm{x}_{t}=f(t)\bm{x}_{t}\mathrm{d}t+g^{2}(t)\nabla_{\bm{x}_{t}}\log q (\bm{x}_{T}=\bm{y}|\bm{x}_{t})+g(t)\mathrm{d}\bm{w}_{t},\quad\bm{x}_{0}\sim q_ {0}=p_{\text{data}},\bm{x}_{T}=\bm{y}.$ (5) $$

The endpoint $\bm{y}$ is not restricted to Gaussian noise as in diffusion models, but instead chosen as informative priors (such as the degraded image in image restoration tasks). Given a starting point $\bm{x}_{0}$, the process in Eqn. (5) also owns an analytic forward transition kernel

$$ $q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})=\mathcal{N}(a_{t}\bm{x}_{T}+b_{t}\bm{x}_{0 },c_{t}^{2}\bm{I}),\quad a_{t}=\frac{\alpha_{t}}{\alpha_{T}}\frac{\text{SNR}_{ T}}{\text{SNR}_{t}},b_{t}=\alpha_{t}(1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}}), c_{t}^{2}=\sigma_{t}^{2}(1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}})$ (6) $$

which forms a diffusion bridge, and $\text{SNR}_{t}=\alpha_{t}^{2}/\sigma_{t}^{2}$ is the signal-to-noise ratio at time $t$. DDBMs show that
the forward process Eqn. (5) is associated with a reverse SDE and a probability flow ODE starting from $\bm{x}_{T}=\bm{y}$:

$$ $\displaystyle\mathrm{d}\bm{x}_{t}$ $\displaystyle=\left[f(t)\bm{x}_{t}-g^{2}(t)\left(\nabla_{\bm{x}_{t}}\log q(\bm {x}_{t}|\bm{x}_{T}=\bm{y})-\nabla_{\bm{x}_{t}}\log q_{tT}(\bm{x}_{T}=\bm{y}| \bm{x}_{t})\right)\right]\mathrm{d}t+g(t)\mathrm{d}\bm{w}_{t},$ (7) $\displaystyle\mathrm{d}\bm{x}_{t}$ $\displaystyle=\left[f(t)\bm{x}_{t}-g^{2}(t)\left(\frac{1}{2}\nabla_{\bm{x}_{t} }\log q(\bm{x}_{t}|\bm{x}_{T}=\bm{y})-\nabla_{\bm{x}_{t}}\log q_{tT}(\bm{x}_{T }=\bm{y}|\bm{x}_{t})\right)\right]\mathrm{d}t.$ (8) $$

They share the same marginal distributions $\{q(\bm{x}_{t}|\bm{x}_{T}=\bm{y})\}_{t=0}^{T}$ with the forward process, where $\bar{\boldsymbol{w}}_{t}$ is the reverse-time Wiener process, $q_{tT}$ is analytically known similar to Eqn. (2), and the only unknown term $\nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{T}=\bm{y})$ is the bridge score function. Denoising bridge score matching (DBSM) is proposed to learn the unknown score term $q(\bm{x}_{t}|\bm{x}_{T}=\bm{y})$ with a parameterized network $\bm{s}_{\theta}(\bm{x}_{t},t,\bm{y})$, by minimizing

$$ $\mathcal{L}_{w}(\theta)=\mathbb{E}_{t}\mathbb{E}_{(\bm{x}_{0},\bm{y})\sim p_{ \text{data}}(\bm{x}_{0},\bm{y})}\mathbb{E}_{\bm{x}_{t}\sim q(\bm{x}_{t}|\bm{x} _{0},\bm{x}_{T}=\bm{y})}\left[w(t)\|\bm{s}_{\theta}(\bm{x}_{t},t,\bm{y})- \nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T}=\bm{y})\|_{2}^{2}\right]$ (9) $$

where $q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T}=\bm{y})$ is the forward transition kernel in Eqn. (6) and $w(t)$ is a positive weighting function. To sample from diffusion bridges with Eqn. (7) and Eqn. (8), DDBMs propose a high-order hybrid sampler that alternately simulates the ODE and SDE steps to enhance the sample quality, inspired by the Heun sampler in diffusion models (Karras et al., ). However, it is not dedicated to diffusion bridges and lacks theoretical insights in developing efficient diffusion samplers.

## 3 Generative Model through Non-Markovian Diffusion Bridges

We start by examining the forward process of the diffusion bridge (Eqn. (5)) on a set of discretized timesteps $0=t_{0}<t_{1}<\dots<t_{N-1}<t_{N}=T$ that will be used for reverse sampling. Since the bridge score $\nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{T})$ only depends on the marginal distribution $q(\bm{x}_{t}|\bm{x}_{T})$, we can construct alternative probabilistic models that induce new sampling procedures while reusing the learned bridge score $\bm{s}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$, as long as they agree on the $N$ marginals $\{q(\bm{x}_{t_{n}}|\bm{x}_{T})\}_{n=0}^{N-1}$.

### 3.1 Non-Markovian Diffusion Bridges as Forward Process

We consider a family of probability distributions $q^{(\rho)}(\bm{x}_{t_{0:N-1}}|\bm{x}_{T})$, controlled by a variance parameter $\rho\in\mathbb{R}^{N-1}$:

$$ $q^{(\rho)}(\bm{x}_{t_{0:N-1}}|\bm{x}_{T})=q_{0}(\bm{x}_{t_{0}})\prod_{n=1}^{N- 1}q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})$ (10) $$

where $q_{0}$ is the data distribution at time 0 and for $1\leq n\leq N-1$

$$ $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})=\mathcal{N}( a_{t_{n}}\bm{x}_{T}+b_{t_{n}}\bm{x}_{0}+\sqrt{c^{2}_{t_{n}}-\rho^{2}_{n}}\frac {\bm{x}_{t_{n+1}}-a_{t_{n+1}}\bm{x}_{T}-b_{t_{n+1}}\bm{x}_{0}}{c_{t_{n+1}}}, \rho_{n}^{2}\bm{I})$ (11) $$

where $\rho_{n}$ is the $n$-th element of $\rho$ satisfying $\rho_{N-1}=c_{t_{N-1}}$, and $a_{t},b_{t},c_{t}$ are terms related to the noise schedule, as defined in the original diffusion bridge (Eqn. (6)). Intuitively, this decreases the variance (noise level) of the bridge while incorporating additional noise components from the last step. Under this construction, we can prove that $q^{(\rho)}$ maintains consistency in marginal distributions with the original forward process $q$ governed by Eqn. (5).

###### Proposition 3.1 (Marginal Preservation, proof in Appendix B.1 ) .

For $0\leq n\leq N-1$, we have $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{T})=q(\bm{x}_{t_{n}}|\bm{x}_{T})$.

The definition of $q^{(\rho)}$ in Eqn. (10) represents the inference process, since it is factorized as the distribution of $\bm{x}_{t_{n}}$ given $\bm{x}_{t_{n+1}}$ at the previous timestep. Conversely, the forward process $q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T})$ can be induced by Bayes’ rule (Appendix C.1). As $\bm{x}_{t_{n+1}}$ in $q^{(\rho)}$ can simultaneously depend on $\bm{x}_{t_{n}}$ and $\bm{x}_{0}$, we refer to it as non-Markovian diffusion bridges, in contrast to Markovian ones (such as Brownian bridges, and the diffusion bridge defined by the forward SDE in Eqn. (5)) which should satisfy $q(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T})=q(\bm{x}_{t_{n+1}}|
\bm{x}_{t_{n}},\bm{x}_{T})$.

###### Proposition 3.1 (Marginal Preservation, proof in Appendix B.1 ) .

### 3.2 Reverse Generative Process and Equivalent Training Objective

Eqn. (10) can be naturally transformed into a parameterized and learnable generative model, by replacing the unknown $\bm{x}_{0}$ in Eqn. (10) with a data predictor $\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$. Intuitively, $\bm{x}_{t}$ on the diffusion bridge is a weighted mixture of $\bm{x}_{T},\bm{x}_{0}$ and some random Gaussian noise according to Eqn. (6), where the weightings $a_{t},b_{t},c_{t}$ are determined by the timestep $t$. The network $\bm{x}_{\theta}$ is trained to recover the clean data $\bm{x}_{0}$ given $\bm{x}_{t},\bm{x}_{T}$ and $t$.

Specifically, we define the generative process starting from $\bm{x}_{T}$ as

$$ $p_{\theta}(\bm{x}_{t_{n}}|\bm{x}_{t_{n+1}},\bm{x}_{T})=\begin{cases}\mathcal{N }(\bm{x}_{\theta}(\bm{x}_{t_{1}},t_{1},\bm{x}_{T}),\rho_{0}^{2}\bm{I}),\quad n =0\\ q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{\theta}(\bm{x}_{t_{n+1}},t_{n+1},\bm{x}_{T}) ,\bm{x}_{t_{n+1}},\bm{x}_{T}),\quad 1\leq n\leq N-1\end{cases}$ (12) $$

and the joint distribution as $p_{\theta}(\bm{x}_{t_{0:N-1}}|\bm{x}_{T})=\prod_{n=0}^{N-1}p_{\theta}(\bm{x}_{
t_{n}}|\bm{x}_{t_{n+1}},\bm{x}_{T})$. To optimize the network parameter $\theta$, we can adopt the common variational inference objective as in DDPMs (Ho et al., ), except that the distributions are conditioned on $\bm{x}_{T}$:

$$ $\mathcal{J}^{(\rho)}(\theta)=\mathbb{E}_{q(\bm{x}_{T})}\mathbb{E}_{q^{(\rho)}( \bm{x}_{t_{0:N-1}}|\bm{x}_{T})}\left[\log q^{(\rho)}(\bm{x}_{t_{1:N-1}}|\bm{x} _{0},\bm{x}_{T})-\log p_{\theta}(\bm{x}_{t_{0:N-1}}|\bm{x}_{T})\right]$ (13) $$

It seems that the DDBM objective $\mathcal{L}_{w}$ in Eqn. (9) is distinct from $\mathcal{J}^{(\rho)}$: respectively, they are defined on continuous and discrete timesteps; they originate from score matching and variational inference; they have different parameterizations of score and data prediction(^1^11The diffusion bridge models are usually parameterized differently from score prediction, but can be converted to score prediction. See Appendix F.1 for details.). However, we show they are equivalent by focusing on the discretized timesteps and transforming the parameterization.

###### Proposition 3.2 (Training Equivalence, proof in Appendix B.2 ) .

For $\rho>\bm{0}$, there exists certain weights $\gamma$ so that $\mathcal{J}^{(\rho)}(\theta)=\mathcal{L}_{\gamma}(\theta)+C$ on the discretized timesteps $\{t_{n}\}_{n=1}^{N}$, where $C$ is a constant irrelevant to $\theta$. Besides, the bridge score predictor $\bm{s}_{\theta}$ in $\mathcal{L}_{\gamma}(\theta)$ has the following relationship with the data predictor $\bm{x}_{\theta}$ in $\mathcal{J}^{(\rho)}(\theta)$:

$$ $\bm{s}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})=-\frac{\bm{x}_{t}-a_{t}\bm{x}_{T}-b_{ t}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})}{c_{t}^{2}}$ (14) $$

Though the weighting $\gamma$ may not precisely match the actual weighting $w$ for training $\bm{s}_{\theta}$, this discrepancy doesn’t affect our utilization of $\bm{s}_{\theta}$ (Appendix C.2). Hence, it is reasonable to reuse the network trained by $\mathcal{L}$ while leveraging various $\rho$ for improved sampling efficiency.

###### Proposition 3.2 (Training Equivalence, proof in Appendix B.2 ) .

## 4 Sampling with Generalized Diffusion Bridges

Now that we have confirmed the rationality and built the theoretical foundations for applying the generalized diffusion bridge $p_{\theta}$ to pretrained DDBMs, a range of inference processes is now at our disposal, controlled by the variance parameter $\rho$. This positions us to explore the resultant sampling procedures and the effects of $\rho$ in pursuit of better and more efficient generation.

### 4.1 Diffusion Bridge Implicit Models

Suppose we sample in reverse time on the discretized timesteps $0=t_{0}<t_{1}<\dots<t_{N-1}<t_{N}=T$. The number $N$ and the schedule of sampling steps can be made independently of the original timesteps on which the bridge model is trained, whether discrete (Liu et al., ) or continuous (Zhou et al., ). According to the generative process of $p_{\theta}$ in Eqn. (12), the updating rule from $t_{n+1}$ to $t_{n}$ is described by

$$ $\bm{x}_{t_{n}}=a_{t_{n}}\bm{x}_{T}+b_{t_{n}}\hat{\bm{x}}_{0}+\sqrt{c^{2}_{t_{n }}-\rho^{2}_{n}}\underbrace{\frac{\bm{x}_{t_{n+1}}-a_{t_{n+1}}\bm{x}_{T}-b_{t_ {n+1}}\hat{\bm{x}}_{0}}{c_{t_{n+1}}}}_{\text{predicted noise }\hat{\bm{ \epsilon}}}+\rho_{n}\bm{\epsilon},\quad\bm{\epsilon}\sim\mathcal{N}(\bm{0},\bm {I})$ (15) $$

where $\hat{\bm{x}}_{0}=\bm{x}_{\theta}(\bm{x}_{t_{n+1}},t_{n+1},\bm{x}_{T})$ denotes the predicted clean data at time 0.

##### Intuition of the Sampling Procedure

Intuitively, the form of Eqn. (15) resembles the forward transition kernel of the diffusion bridge in Eqn. (6) (which can be rewritten as $\bm{x}_{t}=a_{t}\bm{x}_{T}+b_{t}\bm{x}_{0}+c_{t}\bm{\epsilon},\bm{\epsilon}
\sim\mathcal{N}(\bm{0},\bm{I})$). In comparison, $\bm{x}_{0}$ is substituted with the predicted $\hat{\bm{x}}_{0}$, and a portion of the standard Gaussian noise $\bm{\epsilon}$ now stems from the predicted noise $\hat{\bm{\epsilon}}$. The predicted noise $\hat{\bm{\epsilon}}$ is derived from $\bm{x}_{t_{n+1}}$ at the previous timestep and can be expressed by the predicted clean data $\hat{\bm{x}}_{0}$.

**Table 1: Comparison between different diffusion models and diffusion bridge models.**
|  | Diffusion Models | Diffusion Bridge Models |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
|  | DDPM<br>(Ho et al., ) | ScoreSDE<br>(Song et al., ) | DDIM<br>(Song et al., ) | I^2SB<br>(Liu et al., ) | DDBM<br>(Zhou et al., ) | DBIM (Ours) |
| Noise Schedule | VP | Any | Any | VE | Any | Any |
| Timesteps | Discrete | Continuous | Discrete | Discrete | Continuous | Discrete |
| Forward Distribution | $q(\bm{x}_{n}|\bm{x}_{0})$ | $q(\bm{x}_{t}|\bm{x}_{0})$ | $q(\bm{x}_{n}|\bm{x}_{n-1},\bm{x}_{0})$ | $q(\bm{x}_{n}|\bm{x}_{0},\bm{x}_{N})$ | $q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})$ | $q(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T})$ |
| Inference Process | $p_{\theta}(\bm{x}_{n-1}|\bm{x}_{n})$ | SDE/ODE | $p_{\theta}(\bm{x}_{n-1}|\bm{x}_{n})$ | $p_{\theta}(\bm{x}_{n-1}|\bm{x}_{n})$ | SDE/ODE | $p_{\theta}(\bm{x}_{t_{n}}|\bm{x}_{t_{n+1}},\bm{x}_{T})$ |
| Non-Markovian | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ |

##### Effects of the Variance Parameter

We investigate the effects of the variance parameter $\rho$ from the theoretical perspective by considering two extreme cases. Firstly, we note that when $\rho_{n}=\sigma_{t_{n}}\sqrt{1-\frac{\text{SNR}_{t_{n+1}}}{\text{SNR}_{t_{n}}}}$ for each $0\leq n\leq N-1$, the $\bm{x}_{T}$ term in Eqn. (15) is canceled out. In this scenario, the forward process in Eqn. (4.1) becomes a Markovian bridge (see details in Appendix C.1). Besides, the inference process will get rid of $\bm{x}_{T}$ and simplify
to $p_{\theta}(\bm{x}_{t_{n}}|\bm{x}_{t_{n+1}})$, akin to the sampling mechanism in DDPMs (Ho et al., ). Secondly, when $\rho_{n}=0$ for each $0\leq n\leq N-1$, the inference process will be free from random noise and composed of deterministic iterative updates, characteristic of an implicit probabilistic model (Mohamed & Lakshminarayanan, ). Consequently, we name the resulting model
diffusion bridge implicit models (DBIMs), drawing parallels with denoising diffusion implicit models (DDIMs) (Song et al., ). DBIMs serve as the bridge counterpart and extension of DDIMs, as illustrated in Table 1.

When we choose $\rho$ that lies between these two boundary cases,
we can obtain non-Markovian diffusion bridges with intermediate and non-zero stochastic levels. Such bridges may potentially yield superior sample quality. We present detailed ablations in Section 5.1.

Figure: Figure 2: Illustration of the DBIM’s deterministic sampling procedure when $\rho=0$.
Refer to caption: x1.png

##### The Singularity at the Initial Step for Deterministic Sampling

One important aspect to note regarding DBIMs is that its initial step exhibits singularity when $\rho=0$, a property essentially distinct from DDIMs in diffusion models. Specifically, in the initial step we have $t_{n+1}=T$, and $c_{t_{n+1}}$ in the denominator in Eqn. (15) equals 0. This phenomenon can be understood intuitively: given a fixed starting point $\bm{x}_{T}$, the variable $\bm{x}_{t}$ for $t<T$ is typically still stochastically distributed (the marginal $p_{\theta}(\bm{x}_{t}|\bm{x}_{T})$ is not a Dirac distribution). For instance, in inpainting tasks, there should be various plausible complete images corresponding to a fixed masked image. However, a fully deterministic sampling procedure disrupts such stochasticity.

To be theoretically robust, we employ the other boundary choice $\rho_{n}=\sigma_{t_{n}}\sqrt{1-\frac{\text{SNR}_{t_{n+1}}}{\text{SNR}_{t_{n}}}}$ in the initial step(^2^22With this choice, $c_{t_{n+1}}$ in the denominator in Eqn. (15) will be canceled out.), which is aligned with our previous restriction that $\rho_{N-1}=c_{t_{N-1}}$. This will introduce an additional standard Gaussian noise $\bm{\epsilon}$ which we term as the booting noise. It accounts for the stochasticity of the final sample $\bm{x}_{0}$ under a given fixed $\bm{x}_{T}$ and can be viewed as the latent variable. We illustrate the complete DBIM pipeline in Figure 2.

##### Related Work

We discuss related fast sampling and diffusion bridge works in Appendix A. We also discuss some special cases of DBIM and their connection to related works in Appendix C.3.

### 4.2 Connection to Probability Flow ODE

It is intuitive to perceive that the deterministic sampling can be related to solving an ODE. By setting $\rho=0,t_{n+1}=t$ and $t_{n+1}-t_{n}=\Delta t$ in Eqn. (15), the DBIM updating rule can be reorganized as
$\frac{\bm{x}_{t-\Delta t}}{c_{t-\Delta t}}=\frac{\bm{x}_{t}}{c_{t}}+\left(
\frac{a_{t-\Delta t}}{c_{t-\Delta t}}-\frac{a_{t}}{c_{t}}\right)\bm{x}_{T}+
\left(\frac{b_{t-\Delta t}}{c_{t-\Delta t}}-\frac{b_{t}}{c_{t}}\right)\bm{x}_{
\theta}(\bm{x}_{t},t,\bm{x}_{T})$. As $a_{t},b_{t},c_{t}$ are continuous functions of time $t$ defined in Eqn. (6), the ratios $\frac{a_{t}}{c_{t}}$ and $\frac{b_{t}}{c_{t}}$ also remain continuous functions of $t$. Therefore, DBIM ($\rho=0$) can be treated as an Euler discretization of the following ordinary differential equation (ODE):

$$ $\mathrm{d}\left(\frac{\bm{x}_{t}}{c_{t}}\right)=\bm{x}_{T}\mathrm{d}\left( \frac{a_{t}}{c_{t}}\right)+\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})\mathrm{d} \left(\frac{b_{t}}{c_{t}}\right)$ (16) $$

Though it does not resemble a conventional ODE involving $\mathrm{d}t$, the two infinitesimal terms $\mathrm{d}\left(\frac{a_{t}}{c_{t}}\right)$ and $\mathrm{d}\left(\frac{b_{t}}{c_{t}}\right)$ can be expressed with $\mathrm{d}t$ by the chain rule of derivatives. The ODE form also suggests that with a sufficient number of discretization steps, we can reverse the sampling process and obtain encodings of the observed data, which can be useful for interpolation or other downstream tasks.

In DDBMs, the PF-ODE (Eqn. (8)) involving $\mathrm{d}\bm{x}_{t}$ and $\mathrm{d}t$ is proposed and used for deterministic sampling. We reveal in the following proposition that our ODE
in Eqn. (16) can exactly yield the PF-ODE without relying on the advanced Kolmogorov forward (or Fokker-Planck) equation.

###### Proposition 4.1 (Equivalence to Probability Flow ODE, proof in Appendix B.3 ) .

Suppose $\bm{s}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$ is learned as the ground-truth bridge score $\nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{T})$, and $\bm{x}_{\theta}$ is related to $\bm{s}_{\theta}$ through Eqn. (14), then Eqn. (16) can be converted to the PF-ODE (Eqn. (8)) proposed in DDBMs.

Though the conversion from our ODE to the PF-ODE is straightforward, the reverse conversion can be non-trivial and require complex tools such as exponential integrators (Calvo & Palencia, ; Hochbruck et al., ) (Appendix C.4). We highlight our differences from the PF-ODE in DDBMs: (1) Our ODE has a novel form with exceptional neatness. (2) Despite their theoretical equivalence, our ODE describes the evolution of $\frac{\bm{x}_{t}}{c_{t}}$ rather than $\bm{x}_{t}$, and its discretization is performed with respect to $\mathrm{d}\left(\frac{a_{t}}{c_{t}}\right)$ and $\mathrm{d}\left(\frac{b_{t}}{c_{t}}\right)$ instead of $\mathrm{d}t$. (3) Empirically, DBIMs ($\rho=0$) prove significantly more efficient than the Euler discretization of the PF-ODE, thereby accelerating DDBMs by a substantial margin. (4) In contrast to the fully deterministic ODE, DBIMs are capable of various stochastic levels to achieve the best generation quality under the same sampling steps.

###### Proposition 4.1 (Equivalence to Probability Flow ODE, proof in Appendix B.3 ) .

### 4.3 Extension to High-Order Methods

The simplicity and efficiency of our ODE (Eqn. (16)) also inspire novel high-order numerical solvers tailored for DDBMs, potentially bringing faster convergence than the first-order Euler discretization. Specifically, using the time change-of-variable $\lambda_{t}=\log\left(\frac{b_{t}}{c_{t}}\right)=\frac{1}{2}\left(\text{SNR}_{
t}-\text{SNR}_{T}\right)$, the solution of Eqn. (16) from time $t$ to time $s<t$ can be represented as

$$ $\bm{x}_{s}=\frac{c_{s}}{c_{t}}\bm{x}_{t}+\left(a_{s}-\frac{c_{s}}{c_{t}}a_{t} \right)\bm{x}_{T}+c_{s}\int_{\lambda_{t}}^{\lambda_{s}}e^{\lambda}\bm{x}_{ \theta}(\bm{x}_{t_{\lambda}},t_{\lambda},\bm{x}_{T})\mathrm{d}\lambda$ (17) $$

where $t_{\lambda}$ is the inverse function of $\lambda_{t}$. The intractable integral can be approximated by Taylor expansion of $\bm{x}_{\theta}$ and finite difference estimations of high-order derivatives, following well-established numerical methods (Hochbruck & Ostermann, ) and their extensive application in diffusion models (Zhang & Chen, ; Lu et al., ; Gonzalez et al., ). We present the derivations of our high-order solvers in Appendix D, and the detailed algorithm in Appendix E.

## 5 Experiments

Figure: Figure 3: Image translation results on the DIODE-Outdoor dataset with DDBM and DBIM.
Refer to caption: extracted/5978797/figures/diode_small/y_0_crop.jpg

In this section, we show that DBIMs surpass the original sampling procedure of DDBMs by a large margin, in terms of both sample quality and sample efficiency. We also showcase DBIM’s capabilities in latent-space encoding, reconstruction, and interpolation using deterministic sampling. All comparisons between DBIMs and DDBMs are conducted using identically trained models. For DDBMs, we employ their proposed hybrid sampler for sampling. For DBIMs, we control the variance parameter $\rho$ by interpolating between its boundary selections:

$$ $\rho_{n}=\eta\sigma_{t_{n}}\sqrt{1-\frac{\text{SNR}_{t_{n+1}}}{\text{SNR}_{t_{ n}}}},\quad\eta\in[0,1]$ (18) $$

where $\eta=0$ and $\eta=1$ correspond to deterministic sampling and Markovian stochastic sampling.

We conduct experiments including (1) image-to-image translation tasks on Edges$\to$Handbags (Isola et al., ) ($64\times 64$) and DIODE-Outdoor (Vasiljevic et al., ) ($256\times 256$) (2) image restoration task of inpainting on ImageNet (Deng et al., ) ($256\times 256$) with $128\times 128$ center mask.
We report the Fréchet inception distance (FID) (Heusel et al., ) for all experiments, and additionally measure Inception Scores (IS) (Barratt & Sharma, ), Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al., ), Mean Square Error (MSE) (for image-to-image translation) and Classifier Accuracy (CA) (for image inpainting), following previous works (Liu et al., ; Zhou et al., ). The metrics are computed using the complete training set for Edges$\to$Handbags and DIODE-Outdoor, and 10,000 images from validation set for ImageNet. Additional experiment details are provided in Appendix F.

### 5.1 Sample Quality and Efficiency

**Table 2: Quantitative results in the image translation task. ^†Baseline results are taken directly from DDBMs, where they did not report the exact NFE.**
|  |  | Edges$\to$Handbags ($64\times 64$) | DIODE-Outdoor ($256\times 256$) |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | NFE | FID $\downarrow$ | IS $\uparrow$ | LPIPS $\downarrow$ | MSE $\downarrow$ | FID $\downarrow$ | IS $\uparrow$ | LPIPS $\downarrow$ | MSE $\downarrow$ |
| Pix2Pix (Isola et al., ) | 1 | 74.8 | 3.24 | 0.356 | 0.209 | 82.4 | 4.22 | 0.556 | 0.133 |
| DDIB (Su et al., ) | $\geq 40$^† | 186.84 | 2.04 | 0.869 | 1.05 | 242.3 | 4.22 | 0.798 | 0.794 |
| SDEdit (Meng et al., ) | $\geq 40$ | 26.5 | 3.58 | 0.271 | 0.510 | 31.14 | 5.70 | 0.714 | 0.534 |
| Rectified Flow (Liu et al., ) | $\geq 40$ | 25.3 | 2.80 | 0.241 | 0.088 | 77.18 | 5.87 | 0.534 | 0.157 |
| I^2SB (Liu et al., ) | $\geq 40$ | 7.43 | 3.40 | 0.244 | 0.191 | 9.34 | 5.77 | 0.373 | 0.145 |
| DDBM (Zhou et al., ) | 118 | 1.83 | 3.73 | 0.142 | 0.040 | 4.43 | 6.21 | 0.244 | 0.084 |
| DDBM (Zhou et al., ) | 200 | 0.88 | 3.69 | 0.110 | 0.006 | 3.34 | 5.95 | 0.215 | 0.020 |
| DBIM (Ours) | 20 | 1.74 | 3.64 | 0.095 | 0.005 | 4.99 | 6.10 | 0.201 | 0.017 |
| DBIM (Ours) | 100 | 0.89 | 3.62 | 0.100 | 0.006 | 2.57 | 6.06 | 0.198 | 0.018 |

**Table 3: Quantative results in the image restoration task.**
| Inpainting |  | ImageNet ($256\times 256$) |  |
| --- | --- | --- | --- |
| Center ($128\times 128$) | NFE | FID $\downarrow$ | CA $\uparrow$ |
| DDRM (Kawar et al., ) | 20 | 24.4 | 62.1 |
| $\Pi$GDM (Song et al., ) | 100 | 7.3 | 72.6 |
| DDNM (Wang et al., ) | 100 | 15.1 | 55.9 |
| Palette (Saharia et al., ) | 1000 | 6.1 | 63.0 |
| I^2SB (Liu et al., ) | 1000 | 4.9 | 66.1 |
| DDBM (Zhou et al., ) | 500 | 4.27 | 71.8 |
| DBIM  (Ours) | 20 | 4.17 | 72.2 |
| DBIM  (Ours) | 100 | 3.88 | 72.6 |

We present the quantitative results of DBIMs in Table 2 and Table 4, compared with baselines including GAN-based, diffusion-based and bridge-based methods. We set the number of function evaluations (NFEs) of DBIM to 20 and 100 to demonstrate both efficiency at small NFEs and quality at large NFEs. We select $\eta$ from the set [0.0, 0.3, 0.5, 0.8, 1.0] for DBIM and report the best results.

In image translation tasks, DDBM achieves the best sample quality (measured by FID) among the baselines, but requires $\text{NFE}>100$. In contrast, DBIM with only $\text{NFE}=20$ already surpasses all baselines, performing better than or on par with DDBM at $\text{NFE}=118$. When increasing the NFE to 100, DBIM further improves the sample quality and outperforms DDBM with $\text{NFE}=200$ on DIODE-Outdoor. In the more challenging image inpainting task on ImageNet $256\times 256$, the superiority of DBIM is highlighted even further. In particular, DBIM with $\text{NFE}=20$ outperforms all baselines, including DDBM with $\text{NFE}=500$, achieving a 25$\times$ speed-up. With $\text{NFE}=100$, DBIM continues to improve sample quality, reaching a FID lower than 4 for the first time.

The comparison of visual quality is illustrated in Figure 1 and Figure 3, where DBIM produces smoother outputs with significantly fewer noisy artifacts compared to DDBM’s hybrid sampler. Additional samples are provided in Appendix G.

**Table 5: Ablation of the variance parameter controlled by $\eta$ for image translation, measured by FID.**
| Sampler |  | NFE |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | 5 | 10 | 20 | 50 | 100 | 200 | 500 | 5 | 10 | 20 | 50 | 100 | 200 | 500 |  |
|  |  | Image Translation, Edges$\to$Handbags ($64\times 64$) | Image Translation, DIODE-Outdoor ($256\times 256$) |  |  |  |  |  |  |  |  |  |  |  |  |
| $\eta$ | 0.0 | 3.60 | 2.46 | 1.74 | 1.14 | 0.89 | 0.74 | 0.65 | 14.25 | 7.98 | 4.99 | 3.20 | 2.57 | 2.27 | 2.09 |
| 0.3 | 3.70 | 2.63 | 1.93 | 1.26 | 0.99 | 0.85 | 0.74 | 14.44 | 8.18 | 5.12 | 3.31 | 2.69 | 2.33 | 2.13 |  |
| 0.5 | 3.78 | 2.76 | 1.99 | 1.35 | 1.04 | 0.88 | 0.74 | 14.89 | 8.63 | 5.50 | 3.62 | 2.91 | 2.46 | 2.21 |  |
| 0.8 | 3.97 | 3.08 | 2.20 | 1.64 | 1.22 | 1.01 | 0.80 | 16.34 | 10.17 | 6.85 | 4.57 | 3.56 | 2.93 | 2.46 |  |
| 1.0 | 4.28 | 3.50 | 2.64 | 2.03 | 1.50 | 1.19 | 0.92 | 19.09 | 12.53 | 8.80 | 5.97 | 4.53 | 3.63 | 2.85 |  |
| DDBM | 317.22 | 137.15 | 46.74 | 7.79 | 2.40 | 0.88 | 0.53 | 328.33 | 151.93 | 41.03 | 15.19 | 6.54 | 3.34 | 2.26 |  |

Ablation of the Variance Parameter We investigate the impact of the variance parameter $\rho$ (controlled by $\eta$) to identify how the level of stochasticity affects sample quality across various NFEs, as shown in Table 4 and Table 5. For image translation tasks, we consistently observe that employing a deterministic sampler with $\eta=0$ yields superior performance compared to stochastic samplers with $\eta>0$. We attribute it to the characteristics of the datasets, where the target image is highly correlated with and dependent on the condition, resulting in a generative model that lacks diversity. In this case, a straightforward mapping without the involvement of stochasticity is preferred. Conversely, for image inpainting on the more diverse dataset ImageNet 256$\times$256, the parameter $\eta$ exhibits significance across different NFEs. When $\text{NFE}\leq 20$, $\eta=0$ is the optimal choice, with FID steadily increasing as $\eta$ ascends. However, when $\text{NFE}\geq 50$, a relatively large level of stochasticity at $\eta=0.8$ or even $\eta=1$ yields optimal FID. Notably, the FID of $\eta=0$ converges to 3.95 at $\text{NFE}=100$, with no further improvement at larger NFEs, indicating convergence to the ground-truth sample by the corresponding PF-ODE. This observation aligns with diffusion models, where deterministic sampling facilitates rapid convergence, while introducing stochasticity in sampling enhances diversity, ultimately culminating in the highest sample quality when NFE is substantial.

**Table 6: The effects of high-order methods, measured by FID.**
| Sampler | NFE |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 5 | 10 | 20 | 50 | 100 | 5 | 10 | 20 | 50 | 100 | 5 | 10 | 20 | 50 | 100 |  |
|  | Image Translation | Inpainting |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  | Edges$\to$Handbags ($64\times 64$) | DIODE-Outdoor ($256\times 256$) | ImageNet ($256\times 256$) |  |  |  |  |  |  |  |  |  |  |  |  |
| DBIM ($\eta=0$) | 3.60 | 2.46 | 1.74 | 1.14 | 0.89 | 14.25 | 7.98 | 4.99 | 3.20 | 2.57 | 6.47 | 4.62 | 4.17 | 4.00 | 3.95 |
| DBIM (2nd-order) | 3.44 | 2.15 | 1.46 | 0.97 | 0.78 | 13.70 | 7.23 | 4.37 | 2.87 | 2.41 | 5.85 | 4.45 | 4.15 | 4.00 | 3.97 |
| DBIM (3rd-order) | 3.38 | 2.09 | 1.43 | 0.95 | 0.77 | 13.42 | 7.03 | 4.22 | 2.85 | 2.40 | 5.83 | 4.42 | 4.12 | 4.00 | 3.96 |

##### High-Order Methods

We further demonstrate the effects of high-order methods by comparing them to deterministic DBIM, the first-order case. As shown in Table 6, high-order methods consistently improve FID scores in image translation tasks, as well as in inpainting tasks when NFE$<50$, resulting in enhanced generation quality in the low NFE regime. Besides, the 3rd-order variant performs slightly better than the 2nd-order variant. However, in contrast to the numerical solvers in diffusion models, the benefits of high-order extensions are relatively minor in diffusion bridges and less pronounced than the improvement when adjusting $\eta$ from 1 to 0. Nevertheless, high-order DBIMs are significantly more efficient than DDBM’s PF-ODE-based high-order solvers.

As illustrated in Figure 1, our high-order sampler produces images of similar semantic content to the first-order case, using the same booting noise. In contrast, the visual quality is improved with finer textures, resulting in better FID. This indicates that the high-order gradient information from past network outputs benefits the generation quality by adding high-frequency visual details.

### 5.2 Reconstruction and Interpolation

Figure: (a) Encoding/Reconstruction
Refer to caption: x2.png

As discussed in Section 4.2, the deterministic nature of DBIMs at $\eta=0$ and its connection to neural ODEs enable faithful encoding and reconstruction by treating the booting noise as the latent variable. Furthermore, employing spherical linear interpolation in the latent space and subsequently decoding back to the image space allows for semantic image interpolation in image translation and image restoration tasks. These capabilities cannot be achieved by DBIMs with $\eta>0$, or by DDBM’s hybrid sampler which incorporates stochastic steps. We showcase the encoding and decoding results in Figure 4a, indicating that accurate reconstruction is achievable with a sufficient number of sampling steps. We also illustrate the interpolation process in Figure 4b.

## 6 Conclusion

In this work, we introduce diffusion bridge implicit models (DBIMs) for accelerated sampling of DDBMs without extra training. In contrast to DDBM’s continuous-time generation processes, we concentrate on discretized sampling steps and propose a series of generalized diffusion bridge models including non-Markovian variants. The induced sampling procedures serve as bridge counterparts and extensions of DDIMs and are further extended to develop high-order numerical solvers, filling the missing perspectives in the context of diffusion bridges. Experiments on high-resolution datasets and challenging inpainting tasks demonstrate DBIM’s superiority in both the sample quality and sample efficiency, achieving state-of-the-art FID scores with 100 steps and providing up to 25$\times$ acceleration of DDBM’s sampling procedure.

#### Acknowledgments

This work was supported by the National Key Research and Development Program of China (No. 2021ZD0110502), NSFC Projects (Nos. 62061136001, 62106123, 62076147, U19A2081, 61972224, 62106120), Tsinghua Institute for Guo Qiang, and the High Performance Computing Center, Tsinghua University. J.Z is
also supported by the XPlorer Prize.

## Appendix A Related Work

##### Fast Sampling of Diffusion Models

Fast sampling of diffusion models can be classified into training-free and training-based methods. A prevalent training-free fast sampler is the denoising diffusion implicit models (DDIMs) (Song et al., ) that employ alternative non-Markovian generation processes in place of DDPMs, a discrete-time diffusion model. ScoreSDE (Song et al., ) further links discrete-time DDPMs to continuous-time score-based models, unrevealing the generation process to be ordinary and stochastic differential equations (ODEs and SDEs). Subsequent training-free samplers concentrate on developing dedicated numerical solvers to the diffusion ODE or SDE, particularly Heun’s methods (Karras et al., ) and exponential integrators (Zhang & Chen, ; Lu et al., ; Zheng et al., ; Gonzalez et al., ). These methods typically require around 10 steps for high-quality generation. In contrast, training-based methods, particularly adversarial distillation (Sauer et al., ) and consistency distillation (Song et al., ; Kim et al., ), become notable for their ability to achieve high-quality generation with just one or two steps. Developing dedicated differential equation solvers for DDBMs, or exploring bridge distillation methods, would be promising avenues for future research.

##### Diffusion Bridges

Diffusion bridges (De Bortoli et al., ; Liu et al., ; Somnath et al., ; Zhou et al., ; Chen et al., ; Shi et al., ) are a promising generative variant of diffusion models for modeling the transport between two arbitrary distributions. One line of work is the diffusion Schrodinger bridge models (De Bortoli et al., ; Shi et al., ), which solves an entropy-regularized optimal transport problem between two probability distributions. However, their reliance on expensive iterative procedures has limited their application scope, particularly for high-dimensional data. Subsequent works have endeavored to enhance the tractability of the Schrodinger bridge problem by making assumptions such as paired data (Liu et al., ; Somnath et al., ; Chen et al., ). On the other hand, DDBMs (Zhou et al., ) construct diffusion bridges via Doob’s $h$-transform, offering a reverse-time perspective of a diffusion process conditioned on given endpoints. This approach aligns the design spaces and training algorithms of DDBMs closely with those of score-based generative models, leading to state-of-the-art performance in image translation tasks. However, the sampling procedure of DDBMs still relies on inefficient simulations of differential equations, lacking theoretical insights to develop efficient samplers.

## Appendix B Proofs

### B.1 Proof of Proposition 3.1

###### Proof.

Since $q^{(\rho)}$ in Eqn. (10) is factorized as $q^{(\rho)}(\bm{x}_{t_{0:N-1}}|\bm{x}_{T})=q_{0}(\bm{x}_{0})q^{(\rho)}(\bm{x}_{
t_{1:N-1}}|\bm{x}_{0},\bm{x}_{T})$ where $q^{(\rho)}(\bm{x}_{t_{1:N-1}}|\bm{x}_{0},\bm{x}_{T})=\prod_{n=1}^{N-1}q^{(\rho
)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})$, we have $q^{(\rho)}(\bm{x}_{0}|\bm{x}_{T})=q_{0}(\bm{x}_{0})=q(\bm{x}_{0}|\bm{x}_{T})$, which proves the case for $n=0$. For $1\leq n\leq N-1$, we have

$$ $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{T})=\int q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0 },\bm{x}_{T})q^{(\rho)}(\bm{x}_{0}|\bm{x}_{T})\mathrm{d}\bm{x}_{0}$ (19) $$

and

$$ $q(\bm{x}_{t_{n}}|\bm{x}_{T})=\int q(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})q(\bm {x}_{0}|\bm{x}_{T})\mathrm{d}\bm{x}_{0}$ (20) $$

Since $q^{(\rho)}(\bm{x}_{0}|\bm{x}_{T})=q(\bm{x}_{0}|\bm{x}_{T})$, we only need to prove $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{n}}|\bm{x}_{0},
\bm{x}_{T})$.

Firstly, when $n=N-1$, we have $t_{n+1}=T$. Note that $\rho$ is restricted by $\rho_{N-1}=c_{t_{N-1}}$, and Eqn. (11) becomes

$$ $q^{(\rho)}(\bm{x}_{t_{N-1}}|\bm{x}_{0},\bm{x}_{T})=\mathcal{N}(a_{t_{N-1}}\bm{ x}_{T}+b_{t_{N-1}}\bm{x}_{0},c_{t_{N-1}}^{2}\bm{I})$ (21) $$

which is exactly the same as the forward transition kernel of $q$ in Eqn. (6). Therefore, $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{n}}|\bm{x}_{0},
\bm{x}_{T})$ holds for $n=N-1$.

Secondly, suppose $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{n}}|\bm{x}_{0},
\bm{x}_{T})$
holds for $n=k$, we aim to prove that it holds for $n=k-1$. Specifically, $q^{(\rho)}(\bm{x}_{t_{k-1}}|\bm{x}_{0},\bm{x}_{T})$ can be expressed as

$$ $\displaystyle q^{(\rho)}(\bm{x}_{t_{k-1}}|\bm{x}_{0},\bm{x}_{T})$ $\displaystyle=\int q^{(\rho)}(\bm{x}_{t_{k-1}}|\bm{x}_{0},\bm{x}_{t_{k}},\bm{x }_{T})q^{(\rho)}(\bm{x}_{t_{k}}|\bm{x}_{0},\bm{x}_{T})\mathrm{d}\bm{x}_{t_{k}}$ (22) $\displaystyle=\int q^{(\rho)}(\bm{x}_{t_{k-1}}|\bm{x}_{0},\bm{x}_{t_{k}},\bm{x }_{T})q(\bm{x}_{t_{k}}|\bm{x}_{0},\bm{x}_{T})\mathrm{d}\bm{x}_{t_{k}}$ $\displaystyle=\int\mathcal{N}(\bm{x}_{t_{k-1}};\bm{\mu}_{k-1|k},\rho_{k-1}^{2} \bm{I})\mathcal{N}(\bm{x}_{t_{k}};a_{t_{k}}\bm{x}_{T}+b_{t_{k}}\bm{x}_{0},c_{t _{k}}^{2}\bm{I})\mathrm{d}\bm{x}_{t_{k}}$ $$

where

$$ $\bm{\mu}_{k-1|k}=a_{t_{k-1}}\bm{x}_{T}+b_{t_{k-1}}\bm{x}_{0}+\sqrt{c^{2}_{t_{k -1}}-\rho^{2}_{k-1}}\frac{\bm{x}_{t_{k}}-a_{t_{k}}\bm{x}_{T}-b_{t_{k}}\bm{x}_{ 0}}{c_{t_{k}}}$ (23) $$

From (Bishop & Nasrabadi, ) (2.115), $q^{(\rho)}(\bm{x}_{t_{k-1}}|\bm{x}_{0},\bm{x}_{T})$ is a Gaussian, denoted as $\mathcal{N}(\bm{\mu}_{k-1},\Sigma_{k-1})$, where

$$ $\displaystyle\bm{\mu}_{k-1}$ $\displaystyle=a_{t_{k-1}}\bm{x}_{T}+b_{t_{k-1}}\bm{x}_{0}+\sqrt{c^{2}_{t_{k-1} }-\rho^{2}_{k-1}}\frac{a_{t_{k}}\bm{x}_{T}+b_{t_{k}}\bm{x}_{0}-a_{t_{k}}\bm{x} _{T}-b_{t_{k}}\bm{x}_{0}}{c_{t_{k}}}$ (24) $\displaystyle=a_{t_{k-1}}\bm{x}_{T}+b_{t_{k-1}}\bm{x}_{0}$ $$

and

$$ $\displaystyle\Sigma_{k-1}$ $\displaystyle=\rho_{k-1}^{2}\bm{I}+\frac{\sqrt{c_{t_{k-1}}^{2}-\rho^{2}_{k-1}} }{c_{t_{k}}}c_{t_{k}}^{2}\frac{\sqrt{c_{t_{k-1}}^{2}-\rho^{2}_{k-1}}}{c_{t_{k} }}\bm{I}$ (25) $\displaystyle=c_{t_{k-1}}^{2}\bm{I}$ $$

Therefore, $q^{(\rho)}(\bm{x}_{t_{k-1}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{k-1}}|\bm{x}_{
0},\bm{x}_{T})=\mathcal{N}(a_{t_{k-1}}\bm{x}_{T}+b_{t_{k-1}}\bm{x}_{0},c^{2}_{
t_{k-1}}\bm{I})$. By mathematical deduction, $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{n}}|\bm{x}_{0},
\bm{x}_{T})$ holds for every $1\leq n\leq N-1$, which completes the proof.
∎

###### Proof.

### B.2 Proof of Proposition 3.2

###### Proof.

Substituting Eqn. (10) and the joint distribution into Eqn. (13), we have

$$ $\displaystyle\mathcal{J}^{(\rho)}(\theta)$ (26) $\displaystyle=$ $\displaystyle\mathbb{E}_{q(\bm{x}_{T})}\mathbb{E}_{q^{(\rho)}(\bm{x}_{t_{0:N-1 }}|\bm{x}_{T})}\left[\log q^{(\rho)}(\bm{x}_{t_{1:N-1}}|\bm{x}_{0},\bm{x}_{T}) -\log p_{\theta}(\bm{x}_{t_{0:N-1}}|\bm{x}_{T})\right]$ $\displaystyle=$ $\displaystyle\mathbb{E}_{q(\bm{x}_{T})}\mathbb{E}_{q^{(\rho)}(\bm{x}_{t_{0:N-1 }}|\bm{x}_{T})}\left[\sum_{n=1}^{N-1}\log q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0} ,\bm{x}_{t_{n+1}},\bm{x}_{T})-\sum_{n=0}^{N-1}\log p_{\theta}(\bm{x}_{t_{n}}| \bm{x}_{t_{n+1}},\bm{x}_{T})\right]$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{N-1}\mathbb{E}_{q(\bm{x}_{T})}\mathbb{E}_{q^{(\rho)}( \bm{x}_{0},\bm{x}_{t_{n+1}}|\bm{x}_{T})}\left[D_{\mathrm{KL}}(q^{(\rho)}(\bm{x }_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})\;\|\;p_{\theta}(\bm{x}_{t_{n }}|\bm{x}_{t_{n+1}},\bm{x}_{T}))\right]$ $\displaystyle-\mathbb{E}_{q(\bm{x}_{T})}\mathbb{E}_{q^{(\rho)}(\bm{x}_{0},\bm{ x}_{t_{1}}|\bm{x}_{T})}\left[\log p_{\theta}(\bm{x}_{0}|\bm{x}_{t_{1}},\bm{x}_ {T})\right]$ $$

where

$$ $\displaystyle D_{\mathrm{KL}}(q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{ n+1}},\bm{x}_{T})\;\|\;p_{\theta}(\bm{x}_{t_{n}}|\bm{x}_{t_{n+1}},\bm{x}_{T}))$ (27) $\displaystyle=$ $\displaystyle D_{\mathrm{KL}}(q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{ n+1}},\bm{x}_{T})\;\|\;q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{\theta}(\bm{x}_{t_{n+ 1}},t_{n+1},\bm{x}_{T}),\bm{x}_{t_{n+1}},\bm{x}_{T}))$ $\displaystyle=$ $\displaystyle\frac{d_{n}^{2}\|\bm{x}_{\theta}(\bm{x}_{t_{n+1}},t_{n+1},\bm{x}_ {T})-\bm{x}_{0}\|_{2}^{2}}{2\rho_{n}^{2}}$ $$

where we have denoted $d_{n}\coloneqq b_{t_{n}}-\sqrt{c_{t_{n}}^{2}-\rho_{n}^{2}}\frac{b_{t_{n+1}}}{c
_{t_{n+1}}}$. Besides, we have

$$ $\displaystyle\log p_{\theta}(\bm{x}_{0}|\bm{x}_{t_{1}},\bm{x}_{T})$ $\displaystyle=\log q^{(\rho)}(\bm{x}_{0}|\bm{x}_{\theta}(\bm{x}_{t_{1}},t_{1}, \bm{x}_{T}),\bm{x}_{t_{1}},\bm{x}_{T})$ (28) $\displaystyle=\log\mathcal{N}(\bm{x}_{\theta}(\bm{x}_{t_{1}},t_{1},\bm{x}_{T}) ,\rho_{0}^{2}\bm{I})$ $\displaystyle=-\frac{\|\bm{x}_{\theta}(\bm{x}_{t_{1}},t_{1},\bm{x}_{T})-\bm{x} _{0}\|_{2}^{2}}{2\rho_{0}^{2}}+C$ $$

where $C$ is irrelevant to $\theta$. According to Eqn. (6), the conditional score is

$$ $\nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})=-\frac{\bm{x}_{t}- a_{t}\bm{x}_{T}-b_{t}\bm{x}_{0}}{c_{t}^{2}}$ (29) $$

Therefore,

$$ $\displaystyle\|\bm{x}_{\theta}(\bm{x}_{t_{n}},t_{n},\bm{x}_{T})-\bm{x}_{0}\|_{ 2}^{2}$ (30) $\displaystyle=$ $\displaystyle\frac{c^{4}_{t_{n}}}{b^{2}_{t_{n}}}\left\|-\frac{\bm{x}_{t_{n}}-a _{t_{n}}\bm{x}_{T}-b_{t_{n}}\bm{x}_{\theta}(\bm{x}_{t_{n}},t_{n},\bm{x}_{T})}{ c_{t_{n}}^{2}}-\left(-\frac{\bm{x}_{t_{n}}-a_{t_{n}}\bm{x}_{T}-b_{t_{n}}\bm{x} _{0}}{c_{t_{n}}^{2}}\right)\right\|_{2}^{2}$ $\displaystyle=$ $\displaystyle\frac{c^{4}_{t_{n}}}{b^{2}_{t_{n}}}\|\bm{s}_{\theta}(\bm{x}_{t_{n }},t_{n},\bm{x}_{T})-\nabla_{\bm{x}_{t_{n}}}\log q(\bm{x}_{t_{n}}|\bm{x}_{0}, \bm{x}_{T})\|_{2}^{2}$ $$

where $\bm{s}_{\theta}$ is related to $\bm{x}_{\theta}$ by

$$ $\bm{s}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})=-\frac{\bm{x}_{t}-a_{t}\bm{x}_{T}-b_{ t}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})}{c_{t}^{2}}$ (31) $$

Define $d_{0}=1$, the loss $\mathcal{J}^{(\rho)}(\theta)$ is further simplified to

$$ $\displaystyle\mathcal{J}^{(\rho)}(\theta)-C$ (32) $\displaystyle=$ $\displaystyle\sum_{n=0}^{N-1}\mathbb{E}_{q(\bm{x}_{T})q^{(\rho)}(\bm{x}_{0}, \bm{x}_{t_{n+1}}|\bm{x}_{T})}\left[\frac{d_{n}^{2}\|\bm{x}_{\theta}(\bm{x}_{t_ {n+1}},t_{n+1},\bm{x}_{T})-\bm{x}_{0}\|_{2}^{2}}{2\rho_{n}^{2}}\right]$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{N}\frac{d_{n-1}^{2}}{2\rho_{n-1}^{2}}\mathbb{E}_{q( \bm{x}_{T})q(\bm{x}_{0}|\bm{x}_{T})q(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})} \left[\|\bm{x}_{\theta}(\bm{x}_{t_{n}},t_{n},\bm{x}_{T})-\bm{x}_{0}\|_{2}^{2}\right]$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{N}\frac{d_{n-1}^{2}c^{4}_{t_{n}}}{2\rho_{n-1}^{2}b^{2 }_{t_{n}}}\mathbb{E}_{q(\bm{x}_{T})q(\bm{x}_{0}|\bm{x}_{T})q(\bm{x}_{t_{n}}| \bm{x}_{0},\bm{x}_{T})}\left[\|\bm{s}_{\theta}(\bm{x}_{t_{n}},t_{n},\bm{x}_{T} )-\nabla_{\bm{x}_{t_{n}}}\log q(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})\|_{2}^{2 }\right]$ $$

Compared to the training objective of DDBMs in Eqn. (9), $\mathcal{J}^{(\rho)}(\theta)$ is totally equivalent up to a constant, by concentrating on the discretized timesteps $\{t_{n}\}_{n=1}^{N}$, choosing $q(\bm{x}_{T})q(\bm{x}_{0}|\bm{x}_{T})$ as the paired data distribution and using the weighting function $\gamma$ that satisfies $\gamma(t_{n})=\frac{d_{n-1}^{2}c^{4}_{t_{n}}}{2\rho_{n-1}^{2}b^{2}_{t_{n}}}$.
∎

###### Proof.

### B.3 Proof of Proposition 4.1

###### Proof.

We first represent the PF-ODE (Eqn. (8))

$$ $\mathrm{d}\bm{x}_{t}=\left[f(t)\bm{x}_{t}-g^{2}(t)\left(\frac{1}{2}\nabla_{\bm {x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{T})-\nabla_{\bm{x}_{t}}\log q_{tT}(\bm{x}_{T }|\bm{x}_{t})\right)\right]\mathrm{d}t$ (33) $$

with the data predictor $\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$. We replace the bridge score $\nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{T})$ with the network $\bm{s}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$, which is related to $\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$ by Eqn. (14). Besides, $\nabla_{\bm{x}_{t}}\log q_{tT}(\bm{x}_{T}|\bm{x}_{t})$ can be analytically computed as

$$ $\displaystyle\nabla_{\bm{x}_{t}}\log q_{tT}(\bm{x}_{T}|\bm{x}_{t})$ $\displaystyle=\nabla_{\bm{x}_{t}}\log\frac{q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T}) q(\bm{x}_{T}|\bm{x}_{0})}{q(\bm{x}_{t}|\bm{x}_{0})}$ (34) $\displaystyle=\nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})- \nabla_{\bm{x}_{t}}\log q(\bm{x}_{t}|\bm{x}_{0})$ $\displaystyle=-\frac{\bm{x}_{t}-a_{t}\bm{x}_{T}-b_{t}\bm{x}_{0}}{c_{t}^{2}}+ \frac{\bm{x}_{t}-\alpha_{t}\bm{x}_{0}}{\sigma_{t}^{2}}$ $\displaystyle=-\frac{\frac{\text{SNR}_{T}}{\text{SNR}_{t}}(\bm{x}_{t}-\frac{ \alpha_{t}}{\alpha_{T}}\bm{x}_{T})}{\sigma_{t}^{2}(1-\frac{\text{SNR}_{T}}{ \text{SNR}_{t}})}$ $\displaystyle=-\frac{a_{t}(\frac{\alpha_{T}}{\alpha_{t}}\bm{x}_{t}-\bm{x}_{T}) }{c_{t}^{2}}$ $$

Substituting Eqn. (14) and Eqn. (34) into Eqn. (33), the PF-ODE is transformed to

$$ $\displaystyle\mathrm{d}\bm{x}_{t}$ $\displaystyle=\left[f(t)\bm{x}_{t}-g^{2}(t)\left(-\frac{\bm{x}_{t}-a_{t}\bm{x} _{T}-b_{t}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})}{2c_{t}^{2}}+\frac{a_{t}( \frac{\alpha_{T}}{\alpha_{t}}\bm{x}_{t}-\bm{x}_{T})}{c_{t}^{2}}\right)\right] \mathrm{d}t$ (35) $\displaystyle=\left[\left(f(t)+g^{2}(t)\frac{1-2a_{t}\frac{\alpha_{T}}{\alpha_ {t}}}{2c_{t}^{2}}\right)\bm{x}_{t}+g^{2}(t)\frac{a_{t}}{2c_{t}^{2}}\bm{x}_{T}- g^{2}(t)\frac{b_{t}}{2c_{t}^{2}}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})\right ]\mathrm{d}t$ $\displaystyle=\left[\left(f(t)+\frac{g^{2}(t)}{\sigma_{t}^{2}}-\frac{g^{2}(t)} {2c_{t}^{2}}\right)\bm{x}_{t}+g^{2}(t)\frac{a_{t}}{2c_{t}^{2}}\bm{x}_{T}-g^{2} (t)\frac{b_{t}}{2c_{t}^{2}}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})\right] \mathrm{d}t$ $$

On the other hand, the ODE corresponding to DBIMs (Eqn. (16)) can be expanded as

$$ $\frac{\mathrm{d}\bm{x}_{t}}{c_{t}}-\frac{c_{t}^{\prime}}{c_{t}^{2}}\bm{x}_{t} \mathrm{d}t=\left[\left(\frac{a_{t}}{c_{t}}\right)^{\prime}\bm{x}_{T}+\left( \frac{b_{t}}{c_{t}}\right)^{\prime}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T}) \right]\mathrm{d}t$ (36) $$

where we have denoted $(\cdot)^{\prime}\coloneqq\frac{\mathrm{d}(\cdot)}{\mathrm{d}t}$. Further simplification gives

$$ $\mathrm{d}\bm{x}_{t}=\left[\frac{c_{t}^{\prime}}{c_{t}}\bm{x}_{t}+\left(a_{t}^ {\prime}-a_{t}\frac{c_{t}^{\prime}}{c_{t}}\right)\bm{x}_{T}+\left(b_{t}^{ \prime}-b_{t}\frac{c_{t}^{\prime}}{c_{t}}\right)\bm{x}_{\theta}(\bm{x}_{t},t, \bm{x}_{T})\right]\mathrm{d}t$ (37) $$

The coefficients $a_{t},b_{t},c_{t}$ are determined by the noise schedule $\alpha_{t},\sigma_{t}$ in diffusion models. Computing their derivatives will produce terms involving $f(t),g(t)$, which are used to define the forward SDE. As revealed in diffusion models, $f(t),g(t)$ are related to $\alpha_{t},\sigma_{t}$ by $f(t)=\frac{\mathrm{d}\log\alpha_{t}}{\mathrm{d}t},\quad g^{2}(t)=\frac{\mathrm
{d}\sigma_{t}^{2}}{\mathrm{d}t}-2\frac{\mathrm{d}\log\alpha_{t}}{\mathrm{d}t}
\sigma_{t}^{2}$. We can derive the reverse relation of $\alpha_{t},\sigma_{t}$ and $f(t),g(t)$:

$$ $\alpha_{t}=e^{\int_{0}^{t}f(\tau)\mathrm{d}\tau},\quad\sigma_{t}^{2}=\alpha_{t }^{2}\int_{0}^{t}\frac{g^{2}(\tau)}{\alpha_{\tau}^{2}}\mathrm{d}\tau$ (38) $$

which can facilitate subsequent calculation. We first compute the derivative of a common term in $a_{t},b_{t},c_{t}$:

$$ $\left(\frac{1}{\text{SNR}_{t}}\right)^{\prime}=\left(\frac{\sigma_{t}^{2}}{ \alpha_{t}^{2}}\right)^{\prime}=\frac{g^{2}(t)}{\alpha_{t}^{2}}$ (39) $$

For $c_{t}$, since $c_{t}^{2}=\sigma_{t}^{2}(1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}})$, we have

$$ $\frac{c_{t}^{\prime}}{c_{t}}=(\log c_{t})^{\prime}=\frac{1}{2}(\log c_{t}^{2}) ^{\prime}=\frac{1}{2}(\log\sigma_{t}^{2}+\log(1-\frac{\text{SNR}_{T}}{\text{ SNR}_{t}}))^{\prime}$ (40) $$

where

$$ $(\log\sigma_{t}^{2})^{\prime}=(\log\frac{\sigma_{t}^{2}}{\alpha_{t}^{2}})^{ \prime}+(\log\alpha_{t}^{2})^{\prime}=\frac{g^{2}(t)}{\alpha_{t}^{2}}\frac{ \alpha_{t}^{2}}{\sigma_{t}^{2}}+2f(t)=\frac{g^{2}(t)}{\sigma_{t}^{2}}+2f(t)$ (41) $$

and

$$ $(\log(1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}}))^{\prime}=-\frac{\text{SNR}_{T} }{1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}}}\left(\frac{1}{\text{SNR}_{t}}\right )^{\prime}=-\frac{\text{SNR}_{T}}{c_{t}^{2}}\sigma_{t}^{2}\frac{g^{2}(t)}{ \alpha_{t}^{2}}=-\frac{g^{2}(t)}{c_{t}^{2}}\frac{\text{SNR}_{T}}{\text{SNR}_{t}}$ (42) $$

Substituting Eqn. (41) and Eqn. (42) into Eqn. (40), and using the relation $\frac{\text{SNR}_{T}}{\text{SNR}_{t}}=1-\frac{c_{t}^{2}}{\sigma_{t}^{2}}$, we have

$$ $\frac{c_{t}^{\prime}}{c_{t}}=f(t)+\frac{g^{2}(t)}{2\sigma_{t}^{2}}-\frac{g^{2} (t)}{2c_{t}^{2}}\frac{\text{SNR}_{T}}{\text{SNR}_{t}}=f(t)+\frac{g^{2}(t)}{ \sigma_{t}^{2}}-\frac{g^{2}(t)}{2c_{t}^{2}}$ (43) $$

For $a_{t}$, since $a_{t}=\frac{\alpha_{t}}{\alpha_{T}}\frac{\text{SNR}_{T}}{\text{SNR}_{t}}$, we have

$$ $\frac{a_{t}^{\prime}}{a_{t}}=(\log a_{t})^{\prime}=(\log\alpha_{t})^{\prime}+( \log\frac{\text{SNR}_{T}}{\text{SNR}_{t}})^{\prime}=f(t)+\text{SNR}_{t}\frac{g ^{2}(t)}{\alpha_{t}^{2}}=f(t)+\frac{g^{2}(t)}{\sigma_{t}^{2}}$ (44) $$

For $b_{t}$, since $b_{t}=\alpha_{t}(1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}})$, we have

$$ $\frac{b_{t}^{\prime}}{b_{t}}=(\log b_{t})^{\prime}=(\log\alpha_{t})^{\prime}+( \log(1-\frac{\text{SNR}_{T}}{\text{SNR}_{t}}))^{\prime}=f(t)-\frac{g^{2}(t)}{c _{t}^{2}}\frac{\text{SNR}_{T}}{\text{SNR}_{t}}=f(t)+\frac{g^{2}(t)}{\sigma_{t} ^{2}}-\frac{g^{2}(t)}{c_{t}^{2}}$ (45) $$

Therefore,

$$ $a_{t}^{\prime}-a_{t}\frac{c_{t}^{\prime}}{c_{t}}=a_{t}(\frac{a_{t}^{\prime}}{a _{t}}-\frac{c_{t}^{\prime}}{c_{t}})=\frac{g^{2}(t)}{2c_{t}^{2}}a_{t}$ (46) $$

and

$$ $b_{t}^{\prime}-b_{t}\frac{c_{t}^{\prime}}{c_{t}}=b_{t}(\frac{b_{t}^{\prime}}{b _{t}}-\frac{c_{t}^{\prime}}{c_{t}})=-\frac{g^{2}(t)}{2c_{t}^{2}}b_{t}$ (47) $$

Substituting Eqn. (43), Eqn. (46) and Eqn. (47) into the ODE of DBIMs in Eqn. (37), we obtain exactly the PF-ODE in Eqn. (35).
∎

###### Proof.

## Appendix C More Theoretical Discussions

### C.1 Markov Property of the Generalized Diffusion Bridges

We aim to analyze the Markov property of the forward process corresponding to our generalized diffusion bridge in Section 3.1. The forward process of $q^{(\rho)}$ can be induced by Bayes’ rule as

$$ $q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T})=\frac{q^{( \rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})q^{(\rho)}(\bm{x} _{t_{n+1}}|\bm{x}_{0},\bm{x}_{T})}{q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x} _{T})}$ (48) $$

where $q^{(\rho)}(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})$ is the marginal distribution of the diffusion bridge in Eqn. (6), and $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})$ is defined in Eqn. (11) as

$$ $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{t_{n+1}},\bm{x}_{T})=\mathcal{N}( a_{t_{n}}\bm{x}_{T}+b_{t_{n}}\bm{x}_{0}+\sqrt{c^{2}_{t_{n}}-\rho^{2}_{n}}\frac {\bm{x}_{t_{n+1}}-a_{t_{n+1}}\bm{x}_{T}-b_{t_{n+1}}\bm{x}_{0}}{c_{t_{n+1}}}, \rho_{n}^{2}\bm{I}).$ (49) $$

Due to the marginal preservation property (Proposition 3.1), we have $q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{n+1}}|\bm{x}_{
0},\bm{x}_{T})$ and $q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0},\bm{x}_{T})=q(\bm{x}_{t_{n}}|\bm{x}_{0},
\bm{x}_{T})$, where $q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})=\mathcal{N}(a_{t}\bm{x}_{T}+b_{t}\bm{x}_{0
},c_{t}^{2}\bm{I})$ is the forward transition kernel in Eqn. (6). To identify whether $q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T})$ is Markovian, we only need to examine the dependence of $\bm{x}_{t_{n+1}}$ on $\bm{x}_{0}$. To this end, we proceed to derive conditions under which $\nabla_{\bm{x}_{t_{n+1}}}\log q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{T})$ involves terms concerning $\bm{x}_{0}$.

Specifically, $\nabla_{\bm{x}_{t_{n+1}}}\log q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{T})$ can be calculated as

$$ $\displaystyle\nabla_{\bm{x}_{t_{n+1}}}\log q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_ {0},\bm{x}_{T})$ (50) $\displaystyle=$ $\displaystyle\nabla_{\bm{x}_{t_{n+1}}}\log q^{(\rho)}(\bm{x}_{t_{n}}|\bm{x}_{0 },\bm{x}_{t_{n+1}},\bm{x}_{T})+\nabla_{\bm{x}_{t_{n+1}}}\log q^{(\rho)}(\bm{x} _{t_{n+1}}|\bm{x}_{0},\bm{x}_{T})$ $\displaystyle=$ $\displaystyle-\frac{\sqrt{c_{t_{n}}^{2}-\rho_{n}^{2}}(a_{t_{n}}\bm{x}_{T}+b_{t _{n}}\bm{x}_{0}+\sqrt{c^{2}_{t_{n}}-\rho^{2}_{n}}\frac{\bm{x}_{t_{n+1}}-a_{t_{ n+1}}\bm{x}_{T}-b_{t_{n+1}}\bm{x}_{0}}{c_{t_{n+1}}})}{c_{t_{n+1}}\rho_{n}^{2}}$ $\displaystyle-\frac{\bm{x}_{t_{n+1}}-a_{t_{n+1}}\bm{x}_{T}-b_{t_{n+1}}\bm{x}_{ 0}}{c^{2}_{t_{n+1}}}$ $\displaystyle=$ $\displaystyle\frac{b_{t_{n+1}}c_{t_{n}}^{2}-b_{t_{n}}c_{t_{n+1}}\sqrt{c_{t_{n} }^{2}-\rho_{n}^{2}}}{c_{t_{n+1}}^{2}\rho_{n}^{2}}\bm{x}_{0}+C(\bm{x}_{t_{n}}, \bm{x}_{t_{n+1}},\bm{x}_{T})$ $$

where $C(\bm{x}_{t_{n}},\bm{x}_{t_{n+1}},\bm{x}_{T})$ are terms irrelevant to $\bm{x}_{0}$. Therefore,

$$ $\displaystyle q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T} )\text{ is Markovian}$ $\displaystyle\Longleftrightarrow\frac{b_{t_{n+1}}c_{t_{n}}^{2}-b_{t_{n}}c_{t_{ n+1}}\sqrt{c_{t_{n}}^{2}-\rho_{n}^{2}}}{c_{t_{n+1}}^{2}\rho_{n}^{2}}=0$ (51) $\displaystyle\Longleftrightarrow\rho_{n}=\sigma_{t_{n}}\sqrt{1-\frac{\text{SNR }_{t_{n+1}}}{\text{SNR}_{t_{n}}}}$ $$

which is exactly a boundary choice of the variance parameter $\rho$. Under the other boundary choice $\rho_{n}=0$ and intermediate ones satisfying $0<\rho_{n}<\sigma_{t_{n}}\sqrt{1-\frac{\text{SNR}_{t_{n+1}}}{\text{SNR}_{t_{n}
}}}$, the forward process $q^{(\rho)}(\bm{x}_{t_{n+1}}|\bm{x}_{0},\bm{x}_{t_{n}},\bm{x}_{T})$ is non-Markovian.

### C.2 The Insignificance of Loss Weighting in Training

The insignificance of the weighting mismatch in Proposition 3.2 can be interpreted from two aspects. On the one hand, $\mathcal{L}$ consists of independent terms concerning individual timesteps (as long as the network’s parameters are not shared across different $t$), ensuring that the global minimum remains the same as minimizing the loss at each timestep, regardless of the weighting. On the other hand, $\mathcal{L}$ under different weightings are mutually bounded by $\frac{\min_{t}w_{t}}{\max_{t}\gamma_{t}}\mathcal{L}_{\gamma}(\theta)\leq
\mathcal{L}_{w}(\theta)\leq\frac{\max_{t}w_{t}}{\min_{t}\gamma_{t}}\mathcal{L}
_{\gamma}(\theta)$. Besides, it is widely acknowledged that in diffusion models, the weighting corresponding to variational inference may yield superior likelihood but suboptimal sample quality (Ho et al., ; Song et al., ), which is not preferred in practice.

### C.3 Special Cases and Relationship with Prior Works

##### Connection to Flow Matching

When the noise schedule $\alpha_{t}=1,T=1$ and $\sigma_{t}=\sqrt{\beta t}$, the forward process becomes $q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{1})=\mathcal{N}(t\bm{x}_{T}+(1-t)\bm{x}_{0},
\beta t(1-t))$ which is a Brownian bridge. When $\beta\rightarrow 0$, there will be no intermediate noise and the forward process is similar to flow matching (Lipman et al., ). In this limit, the DBIM ($\eta=1$) updating rule from time $t$ to time $s<t$ will become $\bm{x}_{s}=s\bm{x}_{T}+(1-s)\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})=\frac{s}{
t}\bm{x}_{t}+(1-\frac{s}{t})\bm{x}_{\theta}(\bm{x}_{t},t)=\bm{x}_{t}-(t-s)\bm{
v}_{\theta}(\bm{x}_{t},t)$. Here we define $\bm{v}_{\theta}(\bm{x}_{t},t)\coloneqq\frac{\bm{x}_{t}-\bm{x}_{\theta}(\bm{x}_
{t},t)}{t}$ as the velocity function of the probability flow (i.e., the drift of the ODE) in flow matching methods. Therefore, in the flow matching case, DBIM is a simple Euler step of the flow.

##### Connection to DDIM

In the regions where $t$ is small and $\frac{\text{SNR}_{T}}{\text{SNR}_{t}}$ is close to 0, we have $a_{t}\approx 0,b_{t}\approx\alpha_{t},c_{t}\approx\sigma_{t}$. Therefore, the forward process of DDBM in this case is approximately $q(\bm{x}_{t}|\bm{x}_{0},\bm{x}_{T})=\mathcal{N}(\alpha_{t}\bm{x}_{0},\sigma_{t
}^{2}\bm{I})$, which is the forward process of the corresponding diffusion model. Moreover, in this case, the DBIM ($\eta=0$) step is approximately

$$ $\bm{x}_{s}\approx\frac{\sigma_{s}}{\sigma_{t}}\bm{x}_{t}+\sigma_{s}\left(\frac {\alpha_{s}}{\sigma_{s}}-\frac{\alpha_{t}}{\sigma_{t}}\right)\bm{x}_{\theta}( \bm{x}_{t},t,\bm{x}_{T})$ (52) $$

which is exactly DDIM (Song et al., ), except that the data prediction network $\bm{x}_{\theta}$ is dependent on $\bm{x}_{T}$. This indicates that when $t$ is small so that the component of $x_{T}$ in $x_{t}$ is negligible, DBIM recovers DDIM.

##### Connection to Posterior Sampling

The previous work I^2SB (Liu et al., ) also employs diffusion bridges with discrete timesteps, though their noise schedule is restricted to the variance exploding (VE) type with $f(t)=0$ in the forward process. For generation, they adopt a similar approach to DDPM (Ho et al., ) by iterative sampling from the posterior distribution $p_{\theta}(\bm{x}_{n-1}|\bm{x}_{n})$, which is a parameterized and shortened diffusion bridge between the endpoints $\hat{\bm{x}}_{0}=\bm{x}_{\theta}(\bm{x}_{n},t_{n},\bm{x}_{N})$ and $\bm{x}_{n}$. Since the posterior distribution is not conditioned on $\bm{x}_{T}$ (except through the parameterized network), the corresponding forward diffusion bridge is Markovian. Thus, the posterior sampling in I^2SB is a special case of DBIM by setting $\eta=0$ and $f(t)=0$.

### C.4 Perspective of Exponential Integrators

Exponential integrators (Calvo & Palencia, ; Hochbruck et al., ) are widely adopted in recent works concerning fast sampling of diffusion models (Zhang & Chen, ; Zheng et al., ; Gonzalez et al., ). Suppose we have an ODE

$$ $\mathrm{d}\bm{x}_{t}=[a(t)\bm{x}_{t}+b(t)\bm{F}_{\theta}(\bm{x}_{t},t)]\mathrm {d}t$ (53) $$

where $\bm{F}_{\theta}$ is the parameterized prediction function that we want to approximate with Taylor expansion. The usual way of representing its analytic solution $\bm{x}_{t}$ at time $t$ with respect to an initial
condition $\bm{x}_{s}$ at time $s$ is

$$ $\bm{x}_{t}=\bm{x}_{s}+\int_{s}^{t}[a(\tau)\bm{x}_{\tau}+b(\tau)\bm{F}_{\theta} (\bm{x}_{\tau},\tau)]\mathrm{d}\tau$ (54) $$

By approximating the involved integrals in Eqn. (54), we can obtain direct discretizations of Eqn. (53) such as Euler’s method. The key insight of exponential integrators is that, it is often better to utilize the “semi-linear” structure of Eqn. (53) and analytically cancel the linear term $a(t)\bm{x}_{t}$. This way, we can obtain solutions that only involve integrals of $\bm{F}_{\theta}$ and result in lower discretization errors. Specifically, by the “variation-of-constants” formula, the exact solution of Eqn. (53) can be alternatively given by

$$ $\bm{x}_{t}=e^{\int_{s}^{t}a(\tau)\mathrm{d}\tau}\bm{x}_{s}+\int_{s}^{t}e^{\int _{\tau}^{t}a(r)\mathrm{d}r}b(\tau)\bm{F}_{\theta}(\bm{x}_{\tau},\tau)\mathrm{d}\tau$ (55) $$

We can apply this transformation to the PF-ODE in DDBMs. By collecting the linear terms w.r.t. $\bm{x}_{t}$, Eqn. (8) can be rewritten as (already derived in Appendix B.3)

$$ $\mathrm{d}\bm{x}_{t}=\left[\left(f(t)+\frac{g^{2}(t)}{\sigma_{t}^{2}}-\frac{g^ {2}(t)}{2c_{t}^{2}}\right)\bm{x}_{t}+g^{2}(t)\frac{a_{t}}{2c_{t}^{2}}\bm{x}_{T }-g^{2}(t)\frac{b_{t}}{2c_{t}^{2}}\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T}) \right]\mathrm{d}t$ (56) $$

By corresponding it to Eqn. (53), we have

$$ $a(t)=f(t)+\frac{g^{2}(t)}{\sigma_{t}^{2}}-\frac{g^{2}(t)}{2c_{t}^{2}},\quad b_ {1}(t)=g^{2}(t)\frac{a_{t}}{2c_{t}^{2}},\quad b_{2}(t)=-g^{2}(t)\frac{b_{t}}{2 c_{t}^{2}}$ (57) $$

From Eqn. (43), Eqn. (46) and Eqn. (47), we know

$$ $a(t)=\frac{\mathrm{d}\log c_{t}}{\mathrm{d}t},\quad b_{1}(t)=a_{t}\frac{ \mathrm{d}\log(a_{t}/c_{t})}{\mathrm{d}t},\quad b_{2}(t)=b_{t}\frac{\mathrm{d} \log(b_{t}/c_{t})}{\mathrm{d}t}$ (58) $$

Note that these relations are known in advance when converting from our ODE to the PF-ODE. Otherwise, finding them in this inverse conversion will be challenging. The integrals in Eqn. (55) can then be calculated as $\int_{s}^{t}a(\tau)\mathrm{d}\tau=\log c_{t}-\log c_{s}$. Thus

$$ $e^{\int_{s}^{t}a(\tau)\mathrm{d}\tau}=\frac{c_{t}}{c_{s}},\quad e^{\int_{\tau} ^{t}a(r)\mathrm{d}r}=\frac{c_{t}}{c_{\tau}}$ (59) $$

Therefore, the exact solution in Eqn. (55) becomes

$$ $\displaystyle\bm{x}_{t}$ $\displaystyle=\frac{c_{t}}{c_{s}}\bm{x}_{s}+c_{t}\int_{s}^{t}\frac{a_{\tau}}{c _{\tau}}\bm{x}_{T}\mathrm{d}\log\left(\frac{a_{\tau}}{c_{\tau}}\right)+c_{t} \int_{s}^{t}\frac{b_{\tau}}{c_{\tau}}\bm{x}_{\theta}(\bm{x}_{\tau},\tau,\bm{x} _{T})\mathrm{d}\log\left(\frac{b_{\tau}}{c_{\tau}}\right)$ (60) $\displaystyle=\frac{c_{t}}{c_{s}}\bm{x}_{s}+\left(a_{t}-\frac{c_{t}}{c_{s}}a_{ s}\right)\bm{x}_{T}+c_{t}\int_{s}^{t}\frac{b_{\tau}}{c_{\tau}}\bm{x}_{\theta}( \bm{x}_{\tau},\tau,\bm{x}_{T})\mathrm{d}\log\left(\frac{b_{\tau}}{c_{\tau}}\right)$ $$

which is the same as Eqn. (17) after exchanging $s$ and $t$ and changing the time variable in the integral to $\lambda_{t}=\log\left(\frac{b_{t}}{c_{t}}\right)$.

Lastly, we emphasize the advantage of DBIM over employing exponential integrators. First, deriving our ODE via exponential integrators requires the PF-ODE as preliminary. However, the PF-ODE alone cannot handle the singularity at the start point and presents theoretical challenges. Moreover, the conversion process from the PF-ODE to our ODE is intricate, while DBIM retains the overall simplicity. Additionally, DBIM supports varying levels of stochasticity during sampling, unlike the deterministic nature of ODE-based methods. This stochasticity can mitigate sampling errors via the Langevin mechanism (Song et al., ), potentially enhancing the generation quality.

## Appendix D Derivation of Our High-Order Numerical Solvers

High-order solvers of Eqn. (17) can be developed by approximating $\bm{x}_{\theta}$ in the integral with Taylor expansion. Specifically, as
a function of $\lambda$, we have $\bm{x}_{\theta}(\bm{x}_{t_{\lambda}},t_{\lambda},\bm{x}_{T})\approx\bm{x}_{
\theta}(\bm{x}_{t},t,\bm{x}_{T})+\sum_{k=1}^{n}\frac{(\lambda-\lambda_{t})^{k}
}{k!}\bm{x}^{(k)}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$, where $\bm{x}^{(k)}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})=\frac{\mathrm{d}^{k}\bm{x}_{
\theta}(\bm{x}_{t_{\lambda}},t_{\lambda},\bm{x}_{T})}{\mathrm{d}\lambda^{k}}
\Big{|}_{\lambda=\lambda_{t}}$ is the $k$-th order derivative w.r.t. $\lambda$, which can be estimated with finite difference of past network outputs.

##### 2nd-Order Case

With the Taylor expansion $\bm{x}_{\theta}(\bm{x}_{t_{\lambda}},t_{\lambda},\bm{x}_{T})\approx\bm{x}_{
\theta}(\bm{x}_{t},t,\bm{x}_{T})+(\lambda-\lambda_{t})\bm{x}^{(1)}_{\theta}(
\bm{x}_{t},t,\bm{x}_{T})$, we have

$$ $\displaystyle\int_{\lambda_{t}}^{\lambda_{s}}e^{\lambda}\bm{x}_{\theta}(\bm{x} _{t_{\lambda}},t_{\lambda},\bm{x}_{T})\mathrm{d}\lambda$ $\displaystyle\approx\left(\int_{\lambda_{t}}^{\lambda_{s}}e^{\lambda}\mathrm{d }\lambda\right)\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})+\left(\int_{\lambda_{t }}^{\lambda_{s}}(\lambda-\lambda_{t})e^{\lambda}\mathrm{d}\lambda\right)\bm{x} ^{(1)}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$ (61) $\displaystyle\approx e^{\lambda_{s}}\left[(1-e^{-h})\hat{\bm{x}}_{t}+(h-1+e^{- h})\hat{\bm{x}}_{t}^{(1)}\right]$ $$

where we use $\hat{\bm{x}}_{t}$ to denote the network output at time $t$, and $h=\lambda_{s}-\lambda_{t}>0$. Suppose we have used a previous timestep $u$ ($s<t<u$), the first-order derivative can be estimated by

$$ $\hat{\bm{x}}_{t}^{(1)}\approx\frac{\hat{\bm{x}}_{t}-\hat{\bm{x}}_{u}}{h_{1}}, \quad h_{1}=\lambda_{t}-\lambda_{u}$ (62) $$

##### 3rd-Order Case

With the Taylor expansion $\bm{x}_{\theta}(\bm{x}_{t_{\lambda}},t_{\lambda},\bm{x}_{T})\approx\bm{x}_{
\theta}(\bm{x}_{t},t,\bm{x}_{T})+(\lambda-\lambda_{t})\bm{x}^{(1)}_{\theta}(
\bm{x}_{t},t,\bm{x}_{T})+\frac{(\lambda-\lambda_{t})^{2}}{2}\bm{x}^{(2)}_{
\theta}(\bm{x}_{t},t,\bm{x}_{T})$, we have

$$ $\displaystyle\int_{\lambda_{t}}^{\lambda_{s}}e^{\lambda}\bm{x}_{\theta}(\bm{x} _{t_{\lambda}},t_{\lambda},\bm{x}_{T})\mathrm{d}\lambda$ (63) $\displaystyle\approx$ $\displaystyle\left(\int_{\lambda_{t}}^{\lambda_{s}}e^{\lambda}\mathrm{d} \lambda\right)\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})+\left(\int_{\lambda_{t} }^{\lambda_{s}}(\lambda-\lambda_{t})e^{\lambda}\mathrm{d}\lambda\right)\bm{x}^ {(1)}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$ $\displaystyle+\left(\int_{\lambda_{t}}^{\lambda_{s}}\frac{(\lambda-\lambda_{t} )^{2}}{2}e^{\lambda}\mathrm{d}\lambda\right)\bm{x}^{(2)}_{\theta}(\bm{x}_{t},t ,\bm{x}_{T})$ $\displaystyle\approx$ $\displaystyle e^{\lambda_{s}}\left[(1-e^{-h})\hat{\bm{x}}_{t}+(h-1+e^{-h})\hat {\bm{x}}_{t}^{(1)}+\left(\frac{h^{2}}{2}-h+1-e^{-h}\right)\hat{\bm{x}}_{t}^{(2 )}\right]$ $$

Similarly, suppose we have two previous timesteps $u_{1},u_{2}$ ($s<t<u_{1}<u_{2}$), and denote $h_{1}\coloneqq\lambda_{t}-\lambda_{u_{1}},h_{2}\coloneqq\lambda_{u_{1}}-
\lambda_{u_{2}}$, the first-order and second-order derivatives can be estimated by

$$ $\hat{\bm{x}}_{t}^{(1)}\approx\frac{\frac{\hat{\bm{x}}_{t}-\hat{\bm{x}}_{u_{1}} }{h_{1}}(2h_{1}+h_{2})-\frac{\hat{\bm{x}}_{u_{1}}-\hat{\bm{x}}_{u_{2}}}{h_{2}} h_{1}}{h_{1}+h_{2}},\quad\hat{\bm{x}}_{t}^{(2)}\approx 2\frac{\frac{\hat{\bm{x }}_{t}-\hat{\bm{x}}_{u_{1}}}{h_{1}}-\frac{\hat{\bm{x}}_{u_{1}}-\hat{\bm{x}}_{u _{2}}}{h_{2}}}{h_{1}+h_{2}}$ (64) $$

The high-order samplers for DDBMs also theoretically guarantee the order of convergence, similar to those for diffusion models (Zhang & Chen, ; Lu et al., ; Zheng et al., ). We omit the proofs here as they deviate from our main contributions.

## Appendix E Algorithm

Figure: Algorithm 1 DBIM (high-order)

## Appendix F Experiment Details

### F.1 Model Details

DDBMs and DBIMs are assessed using the same trained diffusion bridge models. For image translation tasks, we directly adopt the pretrained checkpoints provided by DDBMs. The data prediction model $\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})$ mentioned in the main text is parameterized by the network $\bm{F}_{\theta}$ as $\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})=c_{\text{skip}}(t)\bm{x}_{t}+c_{\text
{out}}(t)\bm{F}_{\theta}(c_{\text{in}}(t)\bm{x}_{t},c_{\text{noise}}(t),\bm{x}
_{T})$, where

$$ $\displaystyle c_{\text{in}}(t)=\frac{1}{\sqrt{a_{t}^{2}\sigma_{T}^{2}+b_{t}^{2 }\sigma_{0}^{2}+2a_{t}b_{t}\sigma_{0T}+c_{t}}}$ $\displaystyle,\quad c_{\text{out}}(t)=\sqrt{a_{t}^{2}(\sigma_{T}^{2}\sigma_{0} ^{2}-\sigma_{0T}^{2})+\sigma_{0}^{2}c_{t}}c_{\text{in}}(t)$ (65) $\displaystyle c_{\text{skip}}(t)=(b_{t}\sigma_{0}^{2}+a_{t}\sigma_{0T})c^{2}_{ \text{in}}(t)$ $\displaystyle,\quad c_{\text{noise}}(t)=\frac{1}{4}\log t$ $$

and

$$ $\sigma_{0}^{2}=\mbox{Var}[\bm{x}_{0}],\sigma_{T}^{2}=\mbox{Var}[\bm{x}_{T}], \sigma_{0T}=\mbox{Cov}[\bm{x}_{0},\bm{x}_{T}]$ (66) $$

For the image inpainting task on ImageNet 256$\times$256 with 128$\times$128 center mask, DDBMs do not provide available checkpoints. Therefore, we train a new model from scratch using the noise schedule of I^2SB (Liu et al., ). The network is initialized from the pretrained class-conditional diffusion model on ImageNet 256$\times$256 provided by (Dhariwal & Nichol, ), while additionally conditioned on $\bm{x}_{T}$.
The data prediction model in this case is parameterized by the network $\bm{F}_{\theta}$ as $\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})=\bm{x}_{t}-\sigma_{t}\bm{F}_{\theta}(
\bm{x}_{t},t,\bm{x}_{T})$ and trained by minimizing the loss $\mathcal{L}(\theta)=\mathbb{E}_{t,\bm{x}_{0},\bm{x}_{T}}\left[\frac{1}{\sigma_
{t}^{2}}\|\bm{x}_{\theta}(\bm{x}_{t},t,\bm{x}_{T})-\bm{x}_{0}\|_{2}^{2}\right]$. We train the model on 8 NVIDIA A800 GPU cards with a batch size of 256 for 400k iterations, which takes around 19 days.

### F.2 Sampling Details

We elaborate on the sampling configurations of different approaches, including the choice of timesteps $\{t_{i}\}_{i=0}^{N}$ and details of the samplers. In this work, we adopt $t_{\min}=0.0001$ and $t_{\max}=1$ following (Zhou et al., ). For the DDBM baseline, we use the hybrid, high-order Heun sampler proposed in their work with an Euler step ratio of 0.33, which is the best performing configuration for the image-to-image translation task. We use the same timesteps distributed according to EDM (Karras et al., )’s scheduling $(t_{\max}^{1/\rho}+\frac{i}{N}(t_{\min}^{1/\rho}-t_{\max}^{1/\rho}))^{\rho}$, consistent with the official implementation of DDBM.
For DBIM, since the initial sampling step is distinctly forced to be stochastic, we specifically set it to transition from $t_{\max}$ to $t_{\max}-0.0001$, and employ a simple uniformly distributed timestep scheme in $[t_{\min},t_{\max}-0.0001)$ for the remaining timesteps, across all settings. For interpolation experiments, to enhance diversity, we increase the step size of the first step from 0.0001 to 0.01.

### F.3 License

**Table 7: The used datasets, codes and their licenses.**
| Name | URL | Citation | License |
| --- | --- | --- | --- |
| Edges$\rightarrow$Handbags | [https://github.com/junyanz/pytorch-CycleGAN-and-pix2pix](https://github.com/junyanz/pytorch-CycleGAN-and-pix2pix) | Isola et al. () | BSD |
| DIODE-Outdoor | [https://diode-dataset.org/](https://diode-dataset.org/) | Vasiljevic et al. () | MIT |
| ImageNet | [https://www.image-net.org](https://www.image-net.org) | Deng et al. () | $\backslash$ |
| Guided-Diffusion | [https://github.com/openai/guided-diffusion](https://github.com/openai/guided-diffusion) | Dhariwal & Nichol () | MIT |
| I^2SB | [https://github.com/NVlabs/I2SB](https://github.com/NVlabs/I2SB) | Liu et al. () | CC-BY-NC-SA-4.0 |
| DDBM | [https://github.com/alexzhou907/DDBM](https://github.com/alexzhou907/DDBM) | Zhou et al. () | $\backslash$ |

We list the used datasets, codes and their licenses in Table 7.

## Appendix G Additional Samples

Figure: Figure 5: Edges$\rightarrow$Handbags samples on the translation task.
Refer to caption: extracted/5978797/figures/e2h/y_0_crop.jpg

Figure: Figure 6: DIODE-Outdoor samples on the translation task.
Refer to caption: extracted/5978797/figures/diode/y_0.jpg

Figure: Figure 7: ImageNet $256\times 256$ samples on the inpainting task with center $128\times 128$ mask.
Refer to caption: extracted/5978797/figures/inpaint/y_0_crop.jpg

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: Questions:
1. Can you compare your method with I2SB for inpainting with freeform masks and evaluate I2SB on small NFEs?
2. Can you discuss the limitations of your method or provide failure cases? 
3. Can you comment on how much slower high-order methods are than first-order methods regarding inference time? Does the quality improve further for bigger orders like 4?
4. Can you additionally provide inference time for DBIM and other methods to show the inference time acceleration?
5. I suggest the authors consider the other method of DDBM acceleration using consistency distillation and include its discussion in the literature review; see Consistency Diffusion Bridge Models (NeurIPS-2024). 
 
Typos:
1. It looks like in Eq. 7, it should be \bar{w}_{t} instead of w_{t}.
soundness: 3
strengths: 1. The paper proposes fast sampling for the DDBM method, which achieves impressive results for image-to-image translation problems. Enhancing and accelerating DDBM is essential since these models provide a solid alternative to Flow Matching and Schrodinger Bridge methods for image-to-image problems. 
2. The proposed method is training-free and doesn't require any model parameter fine-tuning. Proposition 3.2 establishes equivalence between DDBM and DBIM objectives up to the constant.
3. Experimental results, which prove that deterministic sampler with $\eta = 0$ leads to better results than stochastic samplers with $\eta > 0$ is novel for DDBM, which suggests operating with stochastic samplers and may provide "averaged" or blurry outputs given initial conditions. This result allows further improvement of DDBM with distillation methods for deterministic samplers like consistency distillation (Consistency Models, ICML-2023).
summary: The paper presents a theoretically justified method for fast sampling from DDBMs. The authors introduce non-Markovian diffusion bridges and derive the sampling procedure with the same marginals for discrete time steps. They show that the training objective of the proposed methods is equivalent to the DDBM objective on the discretized timesteps. The method is validated on image-to-image translation problems using edges2handbags and DIODE-Outdoor datasets and inpainting tasks using the ImageNet dataset. The proposed method achieves comparable results with DDBM on NFE > 100 while using only NFE = 20 for these problems regarding FID, LPIPS, and MSE.
weaknesses: In my opinion, the paper doesn't have significant weaknesses. However, some comments should be discussed additionally:
1. I think comparing the best competitor on the inpainting problem according to Table 3 - the I2SB method - is not enough. According to the I2SB paper, this method can repaint the masked region with semantic structures with only 2-10 NFEs for the inpainting problem with freeform masks. I suggest that the authors compare their method with I2SB for the inpainting problem with freeform masks and show the dependence of its quality (LPIPS, FID) on small NFEs (1-20) following setup from I2SB.
2. The idea of incorporating non-Markovian diffusion bridges seems not novel. For example, the I3SB method (Implicit Image-to-Image Schrodinger Bridge for Image Restoration, https://arxiv.org/abs/2403.06069) shares similar ideas and parameterization for I2SB. I suggest the author add the discussion of this paper and the main differences between their method and I3SB.
3. The diversity of the proposed method should be evaluated quantitatively. It is worth showing how small $\eta$ and NFE values degrade diversity compared to stochastic models. I suggest that the authors compute the diversity score from CMDE (Conditional Image Generation with Score-Based Diffusion Models, https://arxiv.org/abs/2111.13606) as in BBDM (BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models, CVPR-2023).

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: 1. Appendix C.1, lines 892-897. Why do you take gradient $\nabla_{x_{t_{n+1}}} \log q^{\rho} (x_{t_{n + 1}} \vert x_0, x_T)$, not $\nabla_{x_{t_{n+1}}} \log q^{\rho} (x_{t_{n + 1}} \vert x_0, x_{t_n}, x_T)$. Also, the eq. 50 is incorrect, while the result (lines 904 - 906) is correct. Please check the derivation.
2. Line 256 - The statement that $x_T$ is cancelled out under the choice of $\rho_n$ seems to be correct, but a link to the proof should be given.
3. Do I understand correctly, that different variants of variance parameter $\rho$ appear only at inference, i.e., training with eq. 13 is agnostic to $\rho$? 
4. You mention that eq. (8) (from DDBM) is equivalent to (16). But on par with (16) you have additional booting noise required by theory. How previous works (e.g., DDBM) manage to avoid the necessity of this noise? 
5. Lines 427 - 428: ‘straightforward mapping without the involvement of stocahsticity’ - but you have booting noise still, yes?
soundness: 3
strengths: The paper novel in the sense that it proposes to adopt the ideas from DDIM (fast sampling, non-markovian diffusion) to the bridge problem, thereby enhancing DDBMs and expanding the practitioner’s toolset for such kind of problems. So, “creative combinations of existing ideas” - is the best description of the presented research. Also, I would like to notice good experimental section.
summary: The paper proposes an extension of DDBM by adopting the ideas from DDIM for the bridge problem (translating one distribution to another with diffusion).
weaknesses: The majority of my concerns are in the "Questions" section. To be honest, I didn’t check Proposition 3.2 - but it seems to be correct. Proposition 4.1 - also not carefully checked by me, but seems to be correct.
What I do not like much about the paper is that some particular formulas and results (see point 2 in my questions) are stated as is, without proof/reference to the proof. It complicates reading and checking the results.
Some other weaknesses:
1. You have "Related works" section (A) in the Appendix. Ok, but I didn't find a reference to this section from the main text. Furthermore, I think it is important to somehow cover related works in the main text (maybe, with some references to the appendix)
2. No source code in the submission. It is important for reproducibility.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. DDBM proposed the pred-x parameterization motivated by the EDM framework to enhance the predictions and derive a set of scaling functions for distribution translation.  Is your work applicable to this extension? If not, what are the challenges?

2. Any intuitions on why a non-Markovian property is preferred?

NIT: some discussions on OT-accelerated diffusions, such as [1,2], may be helpful.  

[1] https://arxiv.org/pdf/2110.11291
[2] https://arxiv.org/pdf/2405.04795
soundness: 3
strengths: 1. Clear derivation to extend DDBM with DDIM properties. 

2. Nice connections to existing literature such as I2SB, flow matching, and exponential integrators.
summary: The authors construct an intriguing process that aligns with the marginal distributions and the well-known diffusion bridge. Building on the success of DDBM in modeling paired distributions at given endpoints, the authors introduce Diffusion Bridge Implicit Models (DBIMs), emphasizing discrete steps over continuous diffusion. Robust empirical techniques are employed in real-world experiments, significantly enhancing prediction performance.
weaknesses: the notation of $q_{tT}$ in Eq.(7) is confusing.


NIT - the literature section should be revised significantly, for example, Schrodinger in line 502: Schr\"odinger, schrodinger in line 509, 600, 606, sde line 521, rosenbrock in line 543, Dpm-solver in line 576, ode in line 644 should have properly capitalized letters.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: I do not have any particular questions. In my opinion, this is a borderline paper which can be accepted if the space permits.
soundness: 3
strengths: - The approach is presented and sufficiently supported by the theoretical derivations.
- The authors support their method by showing that it indeed achieves significant speed-up of the inference on the same setups as were considered in the DDBM paper while preserving the quality.
- The method does not require retraining and is implemented just by a different sampling of the DDBMs models.
summary: The authors consider the denoising diffusion bridge models (DDBMs [1]) recently proposed as a variant of diffusion models useful for interpolating between two paired distributions. The authors propose a way to accelerate this type of diffusion model without retraining them. The proposed method and the paper are mostly analogical to the Denoising Diffusion Implicit Model (DDIM) [2], which is used to accelerate classical diffusion models for generation such as DDPM [3]. In the experimental section, the authors use the same setups and models as in DDBMs and show that the proposed sampling scheme outperforms the sampling scheme of DDBMs.
weaknesses: The general approach of obtaining a new sampling scheme in analogy to DDIM is known and has already been used for a different type of bridge process in SinSR [4], CVPR 2024, used for super-resolution problems. Specifically, the difference is that in SinSR, the bridge process $q(x_t|x_0, x_T)$ for a given low-resolution image $x_{T} = y$ and a given high-resolution image $x_0$ is given by:
$$q(x_t|x_0, x_T) = \mathcal{N}(x_t|x_0 + \eta_t(y-x_0), k^2\eta_t I), $$
and $\eta_t \rightarrow 1$, while in the current paper, the bridge process $q(x_t|x_0, x_T)$ is given by:
$$
q(x_t \mid x_0, x_T) = \mathcal{N}(a_t x_T + b_t x_0, c_t^2 I),
\quad a_t = \frac{\alpha_t}{\alpha_T} \frac{\text{SNR}_T}{\text{SNR}_t},
\quad b_t = \alpha_t \left( 1 - \frac{\text{SNR}_T}{\text{SNR}_t} \right),
\quad c_t^2 = \sigma_t^2 \left( 1 - \frac{\text{SNR}_T}{\text{SNR}_t} \right),
$$
where $SNR_t = \frac{\alpha_t^2}{\sigma_t^2}$ and $c_t^2 \rightarrow 0$ for $t \rightarrow T$.

Thus, there are two differences. The first one is that in SinSR, the process ends in the Gaussian with center $x_T$, while in the current work, the process ends in the delta function. The second is the different parametrization of these processes. To conclude, the methods SynSR & DBIM (proposed here) are not directly equivalent but one can be derived analogously to the other one with some effort.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: See above.
soundness: 3
strengths: I found the paper relatively straightforward to follow. Unsurprisingly, the empirical results support the claims in the paper as DDIM leads to significant improvements in sampling efficiency over vanilla DDPM. Therefore, it is to be expected that similar variants of other stochastic processes could achieve similar gains (See [1]).
summary: The authors present a fast sampling approach to Denoising Diffusion Bridge Models (DDBMs, Zhou et al.) by constructing a DDIM (Song et al.) analogue of DDBMs and hence denote it as Denoising Bridge Implicit Models (DBIMs). The authors empirically demonstrate the advantages of the proposed method on several image translation and restoration benchmarks. The authors also illustrate the application of DBIMs for reconstruction and interpolation using latent space traversal.
weaknesses: Please see below.

**Reinventing the wheel**: My main concern with the paper is that the authors try to reinvent the DDIM framework in the context of a very specific class of stochastic processes (DDBMs in this case) when previous works already define a framework for generalizing over DDIM (as also noted by the authors in the paper in Sec 5.1, higher order methods and Appendix D). More specifically, Pandey et al.  [2] present a generic framework, Conjugate Integrators, for efficient diffusion model sampling of which DDIM and exponential integrators are special cases (See Section 3.1). As a consequence, their framework can be used to enable fast sampling from any pretrained diffusion/flow/interpolant models. Similarly, [1] presents generalized DDIM, which is a special case of Conjugate Integrators and can be applied to diffusion models like CLD or Blurring diffusion models. The main idea of all these methods is to modulate the drift of the reverse diffusion/flow process to enable fast sampling during inference (See also [3] for a similar approach). Since DDBMs also possess a semi-linear drift, extending these generic frameworks to obtain the corresponding DDIM sampler for DDBMs is trivial without delving into the non-markovian formulation of the original DDIM paper.
Consequently, I feel the paper does not offer any new perspective and the flow is very similar to the DDIM paper (defining the non-markovian process followed by setting rho=0 to enable deterministic sampling). Can the authors clarify how their work presents any novel insights compared to the original DDIM paper (or other works mentioned here [1,2]) since this is not obvious to me from the current explanation in the main text?

**Empirical Comparisons**: Although not my main concern, the empirical comparisons seem a bit unfair. For instance, in Table 3, the authors use $\Pi$-GDM and DDRM as comparison baselines, while these methods are task-agnostic and can be adapted for solving different inverse problems using a single pretrained diffusion model prior. However, following Eq. 9, DDBMs/DBIMs require paired data (x_0, y) for training and thus task-specific. Comparing task-specific models with task-agnostic models seems unfair. Moreover, other important baselines based on flow matching [4] or stochastic interpolants [5] are missing, which have advanced quite a lot in terms of sample quality recently [6]. I would recommend updating Table 3 to remove some baselines like DDRM and Pi-GDM and instead compare DBIM with the latest flow matching or interpolant models.

**Dubious claims**: The authors state the following in the Introduction: 
```
Adapting diffusion models to scenarios where a more informative prior naturally exists, such as image translation/restoration, requires cumbersome techniques like adding conditions to the model (Ho & Salimans, 2021; Saharia et al., 2022), modifying the generation pipeline (Meng et al., 2022; Su et al., 2022) or adding extra guidance terms during sampling (Chung et al., 2022; Kawar et al., 2022). These approaches inevitably result in either sub-par performance or slow and resource-intensive inference processes.
```
I agree that task-agnostic models like DPS or $\Pi$-GDM require extra guidance terms in their sampling process, making inference slow due to expensive Jacobian-vector products. However, in my opinion, a slower sampling process (though there is some recent work to mitigate slower sampling in solving inverse problems using diffusion models [7,8]) will likely be acceptable over training new models for each task. Moreover, these methods are more scalable in practice due to their adaptiveness to different restoration tasks while requiring no paired data. On the other hand, task-specific methods will likely require large paired datasets, which can be expensive to acquire. Therefore, certain models are better suited for specific applications (as also implied by the No-Free lunch). Therefore, this claim in the introduction seems misleading to me. It would be great if the authors could update this claim to more accurately reflect the nuances of different approaches, including recent advancements in mitigating slower sampling for task-agnostic models. Moreover, it would be nice to discuss how their method fits into the broader context of these trade-offs and under what scenarios it might be preferable to use DBIM versus a more general, task-agnostic method.

References:

[1] gDDIM: Generalized denoising diffusion implicit models, Zhang et al.

[2] Efficient Integrators for Diffusion Generative Models, Pandey et al.

[3] Bespoke Solvers for Generative Flow Models, Shaul et al.

[4] Flow Matching for Generative Modeling, Lipman et al.

[5] Stochastic Interpolants: A Unifying Framework for Flows and Diffusions, Albergo et al.

[6] SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers, Ma et al.

[7] Accelerating Diffusion Models for Inverse Problems through Shortcut Sampling, Liu et al.

[8] Fast Samplers for Inverse Problems in Iterative Refinement Models, Pandey et al.

Meta-review: This paper proposes a fast sampling method for Denoising Diffusion Bridge Models (DDBMs) which can interpolate two arbitrary paired distributions. Inspired by diffusion bridges, a non-Markovian process is introduced which requires knowledge of $x_0$ at inference time, which is then approximated with its mean. Empirically improved performance was demonstrated. Although some of reviewers have unresolved concerns about novelty and significance, overall the strengths overweigh weaknesses in my opinion, and I'm pleased to recommend acceptance.

Decision: Accept (Poster)