Local MAP Sampling for Diffusion Models
This work addresses the need for more accurate and interpretable solutions in image restoration and scientific inverse problems, offering a novel framework that bridges probabilistic and optimization approaches.
The paper tackled the problem of improving reconstruction accuracy in inverse problems using diffusion models by introducing Local MAP Sampling (LMAPS), which unifies optimization-based methods with probabilistic interpretations and achieved state-of-the-art performance with gains such as ≥2 dB on tasks like motion deblurring and JPEG restoration.
Diffusion Posterior Sampling (DPS) provides a principled Bayesian approach to inverse problems by sampling from $p(x_0 \mid y)$. However, in practice, the goal of inverse problem solving is not to cover the posterior but to recover the most accurate reconstruction, where optimization-based diffusion solvers often excel despite lacking a clear probabilistic foundation. We introduce Local MAP Sampling (LMAPS), a new inference framework that iteratively solving local MAP subproblems along the diffusion trajectory. This perspective clarifies their connection to global MAP estimation and DPS, offering a unified probabilistic interpretation for optimization-based methods. Building on this foundation, we develop practical algorithms with a probabilistically interpretable covariance approximation, a reformulated objective for stability and interpretability, and a gradient approximation for non-differentiable operators. Across a broad set of image restoration and scientific tasks, LMAPS achieves state-of-the-art performance, including $\geq 2$ dB gains on motion deblurring, JPEG restoration, and quantization, and $>1.5$ dB improvements on inverse scattering benchmarks.