Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling

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(y | x), often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, we propose a novel framework called 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 y t , evolving in parallel with the data-space diffusion x t , which enables the derivation of a closed-form posterior p(x t-1 | x t , 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.

• We propose C-DPS, a novel coupled stochastic framework that introduces a parallel diffusion process in the measurement space y t , evolving jointly with the data-space process x t . This formulation enables principled posterior sampling in diffusion models by treating (x, y) as a unified generative process.

• By explicitly constructing a Markov chain over y t , we derive a closed-form posterior transition p(x t-1 | x t , y t-1 ), eliminating the need to approximate or learn the likelihood term p(y | x t ). This allows for direct integration of the measurement model into the sampling procedure, ensuring consistent Bayesian updates at every diffusion step.

• We develop a scalable and efficient sampling algorithm for C-DPS based on a pre-whitened conjugate gradient solver. This matrix-free implementation retains the runtime efficiency of conventional DPS methods, despite the added complexity of coupled data-measurement diffusion.

• We validate C-DPS through extensive experiments on standard benchmarks, including FFHQ [21] and ImageNet [22]. Our method achieves state-of-the-art performance across multiple inverse problem settings-such as inpainting, deblurring, and super-resolution-both qualitatively and quantitatively.

Notation: Scalars are represented by non-bold letters, (e.g., a or A), vectors by bold lowercase letters (e.g., a), and matrices by bold uppercase letters (e.g., A). The real axis is denoted by R. The symbols 0 and I represent the zero vector and the identity matrix, respectively.