Signal Denoising in Non-Gaussian Noise via Diffusion Regularization

When the signal prior is known but the measurement-noise distribution is unknown and non-Gaussian, a key challenge is how to incorporate data-driven noise information into a principled denoising objective while explicitly preserving the signal prior. We develop a noise-domain variational framework for this setting. Specifically, a noise-domain optimization problem is rigorously derived from the finite-dispersion variational formulation of regularization by denoising diffusion process (RED-diff), with the learned noise distribution treated as the RED-diff prior and the signal structure retained explicitly. The resulting objective admits a maximum a posteriori (MAP) interpretation under a normalized effective noise likelihood. This framework is instantiated as Regularization by Denoising with an Atomic-Norm Prior (RED–Atomic) for gridless line spectral estimation, where the atomic norm explicitly encodes spectral sparsity. The corresponding optimization problem is solved by primal–dual splitting with a Toeplitz semidefinite representation. Numerical experiments demonstrate competitive or improved performance across a range of non-Gaussian noise models.