For example, in medical imaging, it is not always possible to obtain fully sampled images of patients as they require long acquisition times.

In many real-world inverse problems, only incomplete measurement data are available for training which can pose a problem for learning a reconstruction function. Indeed, unsupervised learning using a fixed incomplete measurement process is impossible in general, as there is no information in the nullspace of the measurement operator. This limitation can be overcome by using measurements from multiple operators. While this idea has been successfully applied in various applications, a precise characterization of the conditions for learning is still lacking. In this paper, we fill this gap by presenting necessary and sufficient conditions for learning the underlying signal model needed for reconstruction which indicate the interplay between the number of distinct measurement operators, the number of measurements per operator, the dimension of the model and the dimension of the signals. Furthermore, we propose a novel and conceptually simple unsupervised learning loss which only requires access to incomplete measurement data and achieves a performance on par with supervised learning when the sufficient condition is verified. We validate our theoretical bounds and demonstrate the advantages of the proposed unsupervised loss compared to previous methods via a series of experiments on various imaging inverse problems, such as accelerated magnetic resonance imaging, compressed sensing and image inpainting.

We consider the "matrix sensing" problem of recovering a low-rank (or approximately low rank) p.s.d.

Burer-Monteiro type decomposition associated to the low-rank matrix estimation.

With the rise of large radio interferometric telescopes, particularly the SKA, there is a growing demand for computationally efficient image reconstruction techniques. Existing reconstruction methods, such as the CLEAN algorithm or proximal optimisation approaches, are iterative in nature, necessitating a large amount of compute. These methods either provide no uncertainty quantification or require large computational overhead to do so. Learned reconstruction methods have shown promise in providing efficient and high quality reconstruction. In this article we explore the use of generative neural networks that enable efficient approximate sampling of the posterior distribution for high quality reconstructions with uncertainty quantification. Our RI-GAN framework, builds on the regularised conditional generative adversarial network (rcGAN) framework by integrating a gradient U-Net (GU-Net) architecture - a hybrid reconstruction model that embeds the measurement operator directly into the network. This framework uses Wasserstein GANs to improve training stability in combination with regularisation terms that combat mode collapse, which are typical problems for conditional GANs. This approach takes as input the dirty image and the point spread function (PSF) of the observation and provides efficient, high-quality image reconstructions that are robust to varying visibility coverages, generalises to images with an increased dynamic range, and provides informative uncertainty quantification. Our methods provide a significant step toward computationally efficient, scalable, and uncertainty-aware imaging for next-generation radio telescopes.

The success of diffusion models has driven interest in performing conditional sampling via training-free guidance of the denoising process to solve image restoration and other inverse problems. A popular class of methods, based on Diffusion Posterior Sampling (DPS), attempts to approximate the intractable posterior score function directly. In this work, we present a novel expression for the exact posterior score for purely denoising tasks that is tractable in terms of the unconditional score function. We leverage this result to analyze the time-dependent error in the DPS score for denoising tasks and compute step sizes on the fly to minimize the error at each time step. We demonstrate that these step sizes are transferable to related inverse problems such as colorization, random inpainting, and super resolution. Despite its simplicity, this approach is competitive with state-of-the-art techniques and enables sampling with fewer time steps than DPS.