Provable Compressed Sensing with Generative Priors via Langevin Dynamics
This work addresses the theoretical gap in using generative models for inverse problems like compressed sensing, offering a provable method for researchers in signal processing and machine learning, though it is incremental as it builds on existing gradient descent approaches.
The paper tackles the problem of compressed sensing using generative priors by proposing stochastic gradient Langevin dynamics (SGLD) for signal recovery, proving convergence to the true signal under mild assumptions and showing competitive empirical performance compared to standard gradient descent.
Deep generative models have emerged as a powerful class of priors for signals in various inverse problems such as compressed sensing, phase retrieval and super-resolution. Here, we assume an unknown signal to lie in the range of some pre-trained generative model. A popular approach for signal recovery is via gradient descent in the low-dimensional latent space. While gradient descent has achieved good empirical performance, its theoretical behavior is not well understood. In this paper, we introduce the use of stochastic gradient Langevin dynamics (SGLD) for compressed sensing with a generative prior. Under mild assumptions on the generative model, we prove the convergence of SGLD to the true signal. We also demonstrate competitive empirical performance to standard gradient descent.