Inverse Problems and Posterior Sampling with Diffusion Models

An inverse problem asks to recover $x$ from a corrupted/partial observation $y = A(x) + \text{noise}$; a diffusion model’s learned score gives a principled prior over $x$, turning this into posterior sampling from $p(x\mid y) \propto p(y\mid x)\, p(x)$. The catch is that the intermediate noisy states of the reverse process don’t have a tractable likelihood $p(y\mid x_t)$, so the whole family of methods here is about approximating it: diffusion posterior sampling (DPS) approximates it using the current denoised estimate as a point estimate, and Pi-GDM instead approximates the likelihood with a Gaussian, which admits a closed-form correction. Worked through on classic linear inverse problems (colorization, deblurring, inpainting) and less classical ones (separating speech sources, motion tracking/mapping, and even inverse path planning, treated as a sampling-under-constraints problem).

Outline (TODO — flesh out each)

• The inverse-problem (IP) problem statement
• Approximating the conditional score
• Diffusion-based inverse problems; diffusion posterior sampling (DPS)
• Approximating the likelihood with a single sample
• Approximating the likelihood with a Gaussian — Pi-GDM
• Worked examples: colorization, deblurring, inpainting
• Application: source (speech) separation
• Application: motion tracking and mapping
• Application: inverse path planning