Diffusion Posterior Sampling is Computationally Intractable
CoRR(2024)
摘要
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 | x) and a measurement y, and would like to sample from p(x
| 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.
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