David Heurtel-Depeiges

David Heurtel-Depeiges, Blakesley Burkhart, Ruben Ohana et al.

In cosmology, the quest for primordial $B$-modes in cosmic microwave background (CMB) observations has highlighted the critical need for a refined model of the Galactic dust foreground. We investigate diffusion-based modeling of the dust foreground and its interest for component separation. Under the assumption of a Gaussian CMB with known cosmology (or covariance matrix), we show that diffusion models can be trained on examples of dust emission maps such that their sampling process directly coincides with posterior sampling in the context of component separation. We illustrate this on simulated mixtures of dust emission and CMB. We show that common summary statistics (power spectrum, Minkowski functionals) of the components are well recovered by this process. We also introduce a model conditioned by the CMB cosmology that outperforms models trained using a single cosmology on component separation. Such a model will be used in future work for diffusion-based cosmological inference.

David Heurtel-Depeiges, Charles C. Margossian, Ruben Ohana et al.

In recent years, denoising problems have become intertwined with the development of deep generative models. In particular, diffusion models are trained like denoisers, and the distribution they model coincide with denoising priors in the Bayesian picture. However, denoising through diffusion-based posterior sampling requires the noise level and covariance to be known, preventing blind denoising. We overcome this limitation by introducing Gibbs Diffusion (GDiff), a general methodology addressing posterior sampling of both the signal and the noise parameters. Assuming arbitrary parametric Gaussian noise, we develop a Gibbs algorithm that alternates sampling steps from a conditional diffusion model trained to map the signal prior to the family of noise distributions, and a Monte Carlo sampler to infer the noise parameters. Our theoretical analysis highlights potential pitfalls, guides diagnostic usage, and quantifies errors in the Gibbs stationary distribution caused by the diffusion model. We showcase our method for 1) blind denoising of natural images involving colored noises with unknown amplitude and spectral index, and 2) a cosmology problem, namely the analysis of cosmic microwave background data, where Bayesian inference of "noise" parameters means constraining models of the evolution of the Universe.