Bayesian Compressive Sensing Using Generative Models

Generative models, used to create new data resembling a given dataset, have achieved notable success in image reconstruction, especially in scenarios involving incomplete or noisy measurements. In Bayesian compressive sensing and more broadly in Bayesian inverse problems, the posterior inference of unknown quantities plays a crucial role, as it enables the incorporation of prior knowledge, handles ill-posedness, and provides a principled way to quantify uncertainty. However, it encounters challenges with high-dimensional inverse problems and complex prior distributions that are challenging to represent mathematically in traditional Bayesian frameworks. In this paper, we leverage generative models to establish variational inference for posterior mean estimation in Bayesian compressive sensing. The posterior mean estimation is demonstrated to be straightforward and offers theoretical guarantees for Bayesian compressive sensing. Our proposed method, incorporating reparameterization techniques and variational approximations, not only significantly enhances reconstruction quality but also demonstrates robust performance in experiments on compressive sensing and magnetic resonance imaging reconstruction.