The optimization of the latents and parameters of diffusion models with
respect to some differentiable metric defined on the output of the model is a
challenging and complex problem. The sampling for diffusion models is done by
solving either the probability flow ODE or diffusion SDE wherein a neural
network approximates the score function or related quantity, allowing a
numerical ODE/SDE solver to be used. However, na\"ive backpropagation
techniques are memory intensive, requiring the storage of all intermediate
states, and face additional complexity in handling the injected noise from the
diffusion term of the diffusion SDE. We propose a novel method based on the
stochastic adjoint sensitivity method to calculate the gradientwith respect to
the initial noise, conditional information, and model parameters by solving an
additional SDE whose solution is the gradient of the diffusion SDE. We exploit
the unique construction of diffusion SDEs to further simplify the formulation
of the adjoint diffusion SDE and use a change-of-variables to simplify the
solution to an exponentially weighted integral. Using this formulation we
derive a custom solver for the adjoint SDE as well as the simpler adjoint ODE.
The proposed adjoint diffusion solvers can efficiently compute the gradients
for both the probability flow ODE and diffusion SDE for latents and parameters
of the model. Lastly, we demonstrate the effectiveness of the adjoint diffusion
solvers onthe face morphing problem.

扩散模型的优化和参数的梯度计算是一个复杂而具有挑战性的问题，本文提出了一种基于随机伴随灵敏度方法的新方法，用于计算模型的梯度，并演示了该方法在人脸变形问题上的有效性。

AdjointDEIS: 扩散模型的高效梯度

AdjointDEIS: Efficient Gradients for Diffusion Models

We propose the characteristic generator, a novel one-step generative model
that combines the efficiency of sampling in Generative Adversarial Networks
(GANs) with the stable performance of flow-based models. Our model is driven by
characteristics, along which the probability density transport can be described
by ordinary differential equations (ODEs). Specifically, We estimate the
velocity field through nonparametric regression and utilize Euler method to
solve the probability flow ODE, generating a series of discrete approximations
to the characteristics. We then use a deep neural network to fit these
characteristics, ensuring a one-step mapping that effectively pushes the prior
distribution towards the target distribution. In the theoretical aspect, we
analyze the errors in velocity matching, Euler discretization, and
characteristic fitting to establish a non-asymptotic convergence rate for the
characteristic generator in 2-Wasserstein distance. To the best of our
knowledge, this is the first thorough analysis for simulation-free one step
generative models. Additionally, our analysis refines the error analysis of
flow-based generative models in prior works. We apply our method on both
synthetic and real datasets, and the results demonstrate that the
characteristic generator achieves high generation quality with just a single
evaluation of neural network.

我们提出了一种特征生成器，这是一种新颖的单步生成模型，它将生成对抗网络（GANs）中的采样效率与流模型的稳定性相结合。我们的模型驱动着特征，沿着这些特征，可以通过常微分方程（ODEs）来描述概率密度传输。具体而言，我们通过非参数回归估计速度场，并利用欧拉法解决概率流 ODE，生成一系列离散的特征近似。然后，我们使用深度神经网络来拟合这些特征，确保实现了有效地将先验分布推向目标分布的单步映射。在理论方面，我们分析了速度匹配、欧拉离散化和特征拟合中的误差，建立了特征生成器在 2-Wasserstein 距离中的非渐近收敛速率。据我们所知，这是首次对无模拟单步生成模型进行全面分析。此外，我们的分析还改进了先前关于流模型生成模型的误差分析。我们将我们的方法应用于合成数据集和真实数据集，结果表明，仅通过一次神经网络评估即可实现出色的生成质量。

可证明的一步生成的特征学习

Characteristic Learning for Provable One Step Generation

Diffusion-based generative models excel in unconditional generation, as well
as on applied tasks such as image editing and restoration. The success of
diffusion models lies in the iterative nature of diffusion: diffusion breaks
down the complex process of mapping noise to data into a sequence of simple
denoising tasks. Moreover, we are able to exert fine-grained control over the
generation process by injecting guidance terms into each denoising step.
However, the iterative process is also computationally intensive, often taking
from tens up to thousands of function evaluations. Although consistency
trajectory models (CTMs) enable traversal between any time points along the
probability flow ODE (PFODE) and score inference with a single function
evaluation, CTMs only allow translation from Gaussian noise to data. Thus, this
work aims to unlock the full potential of CTMs by proposing generalized CTMs
(GCTMs), which translate between arbitrary distributions via ODEs. We discuss
the design space of GCTMs and demonstrate their efficacy in various image
manipulation tasks such as image-to-image translation, restoration, and
editing. Code: https://github.com/1202kbs/GCTM

本文介绍了扩散式生成模型的成功之处以及其与一般 CTMs 和 GCTMs 的关系，并证明了 GCTMs 在图像操纵任务中的有效性。

图像操作的一般性一致轨迹模型

Generalized Consistency Trajectory Models for Image Manipulation

Score-based generative models are a popular class of generative modelling
techniques relying on stochastic differential equations (SDE). From their
inception, it was realized that it was also possible to perform generation
using ordinary differential equations (ODE) rather than SDE. This led to the
introduction of the probability flow ODE approach and denoising diffusion
implicit models. Flow matching methods have recently further extended these
ODE-based approaches and approximate a flow between two arbitrary probability
distributions. Previous work derived bounds on the approximation error of
diffusion models under the stochastic sampling regime, given assumptions on the
$L^2$ loss. We present error bounds for the flow matching procedure using fully
deterministic sampling, assuming an $L^2$ bound on the approximation error and
a certain regularity condition on the data distributions.

本文提出了一种使用完全确定性采样的流匹配过程，该过程利用概率流 ODE 方法和去噪扩散隐式模型，为基于 ODE 的生成模型提供了误差界限。

流匹配方法的误差界

Error Bounds for Flow Matching Methods

Diffusion models have recently outperformed alternative approaches to model
the distribution of natural images, such as GANs. Such diffusion models allow
for deterministic sampling via the probability flow ODE, giving rise to a
latent space and an encoder map. While having important practical applications,
such as estimation of the likelihood, the theoretical properties of this map
are not yet fully understood. In the present work, we partially address this
question for the popular case of the VP SDE (DDPM) approach. We show that,
perhaps surprisingly, the DDPM encoder map coincides with the optimal transport
map for common distributions; we support this claim theoretically and by
extensive numerical experiments.

研究拟合自然图像分布的传播模型中，VP SDE 方法的编码器映射为常规分布的最优传输映射。