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 computationally intractable: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which every algorithm takes superpolynomial time, even though 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：扩散模型是一种学习并采样分布 $p(x)$ 的非常有效的方法。在后验采样中，我们同时给定一个测量模型 $p(y \\mid x)$ 和一个测量值 $y$，目标是采样 $p(x \\mid y)$。后验采样在图像修复、超分辨率、MRI重建等任务中具有重要应用，因此近期已有若干研究提出了启发式算法来近似实现；但尚未有任何算法被证明能在多项式时间内收敛至正确分布。本文证明后验采样在计算上是难解的：基于密码学中最基本的假设——单向函数存在——存在某些实例使得所有算法均需要超多项式时间，尽管无条件采样已被证明是快速的。我们还表明，在更强的合理假设下（即存在需要指数时间才能求逆的单向函数），指数时间的拒绝采样算法本质上是最优的。