Diffusion models are a remarkably effective way of learning and sampling from a distribution $p(x)$. In posterior sampling, one is also given a measurement model $p(y \mid x)$ and a measurement $y$, and would like to sample from $p(x \mid y)$. Posterior sampling is useful for tasks such as inpainting, super-resolution, and MRI reconstruction, so a number of recent works have given algorithms to heuristically approximate it; but none are known to converge to the correct distribution in polynomial time. In this paper we show that posterior sampling is \emph{computationally intractable}: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which \emph{every} algorithm takes superpolynomial time, even though \emph{unconditional} sampling is provably fast. We also show that the exponential-time rejection sampling algorithm is essentially optimal under the stronger plausible assumption that there are one-way functions that take exponential time to invert.

翻译：扩散模型是从分布 $p(x)$ 中学习和采样的极其有效方法。在后验采样中，我们还需给定测量模型 $p(y \mid x)$ 和测量值 $y$，并希望从 $p(x \mid y)$ 中采样。后验采样在图像修复、超分辨率和MRI重建等任务中非常有用，因此近期许多研究提出了启发式近似算法；但尚无已知算法能在多项式时间内收敛到正确分布。本文证明后验采样在计算上是难解的：在密码学最基本假设（即单向函数存在）下，存在一些实例，其中*所有*算法都需要超多项式时间，尽管*无条件*采样已被证明可以快速完成。我们还证明，在更强的合理假设（即存在需要指数时间才能求逆的单向函数）下，指数时间拒绝采样算法本质上是最优的。