Diffusion Posterior Sampling is Computationally Intractable

• URL: http://arxiv.org/abs/2402.12727v1
• Date: Tue, 20 Feb 2024 05:28:13 GMT
• Title: Diffusion Posterior Sampling is Computationally Intractable
• Authors: Shivam Gupta, Ajil Jalal, Aditya Parulekar, Eric Price, Zhiyang Xun
• Abstract summary: Posterior sampling is useful for tasks such as inpainting, super-resolution, and MRI reconstruction. We show that posterior sampling is emphcomputationally intractable: under the most basic assumption in cryptography, that one-way functions exist. We also show that the exponential-time rejection sampling is essentially optimal under the stronger plausible assumption that there are one-way functions that take exponential time to invert.
• Score: 9.483130965295324
• License: http://creativecommons.org/licenses/by/4.0/
• 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 \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.

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