We propose a conditional stochastic interpolation (CSI) approach to learning
conditional distributions. CSI learns probability flow equations or stochastic
differential equations that transport a reference distribution to the target
conditional distribution. This is achieved by first learning the drift function
and the conditional score function based on conditional stochastic
interpolation, which are then used to construct a deterministic process
governed by an ordinary differential equation or a diffusion process for
conditional sampling. In our proposed CSI model, we incorporate an adaptive
diffusion term to address the instability issues arising during the training
process. We provide explicit forms of the conditional score function and the
drift function in terms of conditional expectations under mild conditions,
which naturally lead to an nonparametric regression approach to estimating
these functions. Furthermore, we establish non-asymptotic error bounds for
learning the target conditional distribution via conditional stochastic
interpolation in terms of KL divergence, taking into account the neural network
approximation error. We illustrate the application of CSI on image generation
using a benchmark image dataset.

条件随机插值 (CSI) 方法用于学习条件分布，通过学习漂移函数、条件评分函数，构建一个由普通微分方程或扩散过程驱动的确定性过程来进行条件采样，并在训练过程中引入自适应扩散项以解决不稳定性问题。我们提供了条件评分函数和漂移函数的明确形式，并建立了通过条件随机插值学习目标条件分布的非渐近误差边界，考虑了神经网络的近似误差。通过在基准图像数据集上生成图像，我们展示了 CSI 方法的应用。

生成性学习的条件随机插值

Conditional Stochastic Interpolation for Generative Learning

Diffusion models are a class of probabilistic generative models that have
been widely used as a prior for image processing tasks like text conditional
generation and inpainting. We demonstrate that these models can be adapted to
make predictions and provide uncertainty quantification for chaotic dynamical
systems. In these applications, diffusion models can implicitly represent
knowledge about outliers and extreme events; however, querying that knowledge
through conditional sampling or measuring probabilities is surprisingly
difficult. Existing methods for conditional sampling at inference time seek
mainly to enforce the constraints, which is insufficient to match the
statistics of the distribution or compute the probability of the chosen events.
To achieve these ends, optimally one would use the conditional score function,
but its computation is typically intractable. In this work, we develop a
probabilistic approximation scheme for the conditional score function which
provably converges to the true distribution as the noise level decreases. With
this scheme we are able to sample conditionally on nonlinear userdefined events
at inference time, and matches data statistics even when sampling from the
tails of the distribution.

通过开发一种概率近似方案，使扩散模型能够在推理时间内有条件地采样非线性用户定义事件，从而匹配数据统计，即使从分布的尾部进行采样。

物理动力系统扩散模型中的用户定义事件采样与不确定性量化

User-defined Event Sampling and Uncertainty Quantification in Diffusion  Models for Physical Dynamical Systems

Diffusion models are powerful, but they require a lot of time and data to
train. We propose Patch Diffusion, a generic patch-wise training framework, to
significantly reduce the training time costs while improving data efficiency,
which thus helps democratize diffusion model training to broader users. At the
core of our innovations is a new conditional score function at the patch level,
where the patch location in the original image is included as additional
coordinate channels, while the patch size is randomized and diversified
throughout training to encode the cross-region dependency at multiple scales.
Sampling with our method is as easy as in the original diffusion model. Through
Patch Diffusion, we could achieve $\mathbf{\ge 2\times}$ faster training, while
maintaining comparable or better generation quality. Patch Diffusion meanwhile
improves the performance of diffusion models trained on relatively small
datasets, $e.g.$, as few as 5,000 images to train from scratch. We achieve
state-of-the-art FID scores 1.77 on CelebA-64$\times$64 and 1.93 on
AFHQv2-Wild-64$\times$64. We will share our code and pre-trained models soon.

提出了一种名为 Patch Diffusion 的通用 Patch-wise 训练框架，其核心创新是新的条件分数函数，它在原始图像的 Patch 级别包括 Patch 位置作为附加坐标通道，并且通过训练在多个尺度上对 Patch 大小进行随机和多样化，从而达到显著减少训练时间成本同时提高数据效率以帮助更广泛的用户学习扩散模型。