Diffusion Posterior Sampling is Computationally Intractable

TL;DR

This paper proves that sampling from the posterior distribution in diffusion models is computationally intractable under standard cryptographic assumptions, highlighting fundamental limitations of current heuristic algorithms.

Contribution

It establishes the computational hardness of posterior sampling in diffusion models, showing no polynomial-time algorithms can reliably perform this task under common cryptographic assumptions.

Findings

Posterior sampling is computationally intractable under cryptographic assumptions.

Unconditional sampling can be fast, but posterior sampling remains hard.

Rejection sampling is near-optimal assuming certain cryptographic conjectures.

Abstract

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.