Proximal-Based Generative Modeling for Bayesian Inverse Problems
This work provides a novel solution to the likelihood score intractability in score-based diffusion models for inverse problems, benefiting applications in imaging and scientific computing.
The paper proposes a proximal-based generative modeling (PGM) framework for Bayesian inverse problems that avoids explicit likelihood evaluation by leveraging a theoretical connection between diffusion processes and Moreau-Yosida regularization. PGM achieves state-of-the-art reconstruction quality and faster sampling, eliminating early-stopping bias.
Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.