Consistency Models Improve Diffusion Inverse Solvers

Diffusion inverse solvers (DIS) aim to find an image $x$ that lives on the diffusion prior while satisfying the constraint $f(x) = y$, given an operator $f(.)$ and measurement $y$. Most non-linear DIS use posterior mean $\hat{x}{0|t}=\mathbb{E}[x0|xt]$ to evaluate $f(.)$ and minimize the distance $||f(\hat{x}{0|t})-y||^2$. Previous works show that posterior mean-based distance is biased; instead, posterior sample $x{0|t}\sim p{\theta}(x0|xt)$ promises a better candidate. In this paper, we first clarify when is posterior sample better: $1)$ When $f(.)$ is linear, the distance with posterior mean is as good as single posterior sample, thus preferable as it does not require Monte Carlo; $2)$ When $f(.)$ is non-linear, the distance using posterior sample is better. As previous approximations to posterior sample do not look like a real image, we propose to use consistency model (CM) as a high quality approximation. In addition, we propose a new family of DIS using pure CM. Empirically, we show that replacing posterior mean by CM improves DIS performance on non-linear $f(.)$ (e.g. semantic segmentation, image captioning). Further, our pure CM inversion works well for both linear and non-linear $f(.)$.