Diffusion models are one of the most important families of deep generative
models. In this note, we derive a quantitative upper bound on the Wasserstein
distance between the data-generating distribution and the distribution learned
by a diffusion model. Unlike previous works in this field, our result does not
make assumptions on the learned score function. Moreover, our bound holds for
arbitrary data-generating distributions on bounded instance spaces, even those
without a density w.r.t. the Lebesgue measure, and the upper bound does not
suffer from exponential dependencies. Our main result builds upon the recent
work of Mbacke et al. (2023) and our proofs are elementary.

我们在这篇论文中，对扩散模型和数据生成分布之间的 Wasserstein 距离进行了定量上界的推导，该结果不依赖于数据生成分布的得分函数，并且适用于具有有界实例空间的任意数据生成分布，即使这些分布对勒贝格测度没有密度，而且上界不会受到指数依赖的影响。

关于去噪扩散概率模型的收敛性的注意事项

A Note on the Convergence of Denoising Diffusion Probabilistic Models

Generative modeling is a widely-used machine learning method with various
applications in scientific and industrial fields. Its primary objective is to
simulate new examples drawn from an unknown distribution given training data
while ensuring diversity and avoiding replication of examples from the training
data.
This paper presents theoretical insights into training a generative model
with two properties: (i) the error of replacing the true data-generating
distribution with the trained data-generating distribution should optimally
converge to zero as the sample size approaches infinity, and (ii) the trained
data-generating distribution should be far enough from any distribution
replicating examples in the training data.
We provide non-asymptotic results in the form of finite sample risk bounds
that quantify these properties and depend on relevant parameters such as sample
size, the dimension of the ambient space, and the dimension of the latent
space. Our results are applicable to general integral probability metrics used
to quantify errors in probability distribution spaces, with the Wasserstein-$1$
distance being the central example. We also include numerical examples to
illustrate our theoretical findings.

使用生成建模方法，本文提供了关于生成模型训练的理论洞察力，重点强调数据生成分布与训练数据生成分布之间的误差应随着样本量趋近无穷而趋近于零，并确保训练数据生成分布与任何复制训练数据样本的分布之间足够远。

最大偏离经验分布的保证最优生成建模

Guaranteed Optimal Generative Modeling with Maximum Deviation from the  Empirical Distribution

We derive, in the classical framework of Bayesian sensitivity analysis,
optimal lower and upper bounds on posterior values obtained from Bayesian
models that exactly capture an arbitrarily large number of finite-dimensional
marginals of the data-generating distribution and/or that are as close as
desired to the data-generating distribution in the Prokhorov or total variation
metrics; these bounds show that such models may still make the largest possible
prediction error after conditioning on an arbitrarily large number of sample
data measured at finite precision. These results are obtained through the
development of a reduction calculus for optimization problems over measures on
spaces of measures. We use this calculus to investigate the mechanisms that
generate brittleness/robustness and, in particular, we observe that learning
and robustness are antagonistic properties. It is now well understood that the
numerical resolution of PDEs requires the satisfaction of specific stability
conditions. Is there a missing stability condition for using Bayesian inference
in a continuous world under finite information?

本文使用降维算法对贝叶斯灵敏度分析中的优化问题进行了探究，结果发现贝叶斯模型在精度固定的有限采样数据的基础上依然可能出现最大可能的预测误差，且学习和稳健性是有冲突的。该研究还探讨了在使用贝叶斯推断的连续世界中是否存在缺失的稳定性条件。