Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling

2025-10-08

AI Summary

Existing diffusion-based inverse problem solvers either rely on heuristic projection steps to enforce measurement consistency or approximate the likelihood $p(y|x)$, leading to artifacts and instability under high noise. This work proposes a coupled diffusion posterior sampling framework jointly operating in data and measurement spaces: it introduces, for the first time, a forward diffusion process explicitly defined in the measurement space and rigorously couples it with the data-space dynamics, yielding a closed-form posterior distribution—thus eliminating likelihood approximation and manual hyperparameter tuning. Bayesian-consistent recursive sampling is achieved via a bidirectional synchronized stochastic process, with denoising score matching jointly optimizing trajectories in both spaces. Evaluated across multiple inverse problem benchmarks, the method significantly improves reconstruction quality and robustness under high noise, consistently outperforming state-of-the-art approaches in both qualitative and quantitative metrics.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Inverse problems, where the goal is to recover an unknown signal from noisy or incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-based techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood $p(oldsymbol{y} mid oldsymbol{x})$, often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, we propose a novel framework called emph{coupled data and measurement space diffusion posterior sampling} (C-DPS), which eliminates the need for constraint tuning or likelihood approximation. C-DPS introduces a forward stochastic process in the measurement space ${oldsymbol{y}_t}$, evolving in parallel with the data-space diffusion ${oldsymbol{x}_t}$, which enables the derivation of a closed-form posterior $p(oldsymbol{x}_{t-1} mid oldsymbol{x}_t, oldsymbol{y}_{t-1})$. This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantitatively, across multiple inverse problem benchmarks.

Problem

• Solving inverse problems with noisy incomplete measurements
• Eliminating constraint tuning and likelihood approximation needs
• Enhancing diffusion posterior sampling accuracy and stability

Innovation

• Coupled data and measurement space diffusion process
• Closed-form posterior distribution for recursive sampling
• Eliminates constraint tuning and likelihood approximation