In practice, when training denoisers, we sample multiple noisy samples for each clean image and minimize the Mean Squared Error (MSE) loss for recovering the clean image.

In practice, when training denoisers, we sample multiple noisy samples for each clean image and minimize the Mean Squared Error (MSE) loss for recovering the clean image.

Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denois-ers instead of the proximal operator or the gradient-descent step within proximal algorithms.

Many important problems in science and engineering involve inferring a signal from noisy and/or incomplete observations, where the observation process is known. Historically, this problem has been tackled using hand-crafted regularization (e.g., sparsity, total-variation) to obtain meaningful estimates. Recent data-driven methods often offer better solutions by directly learning a solver from examples of ground-truth signals and associated observations. However, in many real-world applications, obtaining ground-truth references for training is expensive or impossible. Self-supervised learning methods offer a promising alternative by learning a solver from measurement data alone, bypassing the need for ground-truth references. This manuscript provides a comprehensive summary of different self-supervised methods for inverse problems, with a special emphasis on their theoretical underpinnings, and presents practical applications in imaging inverse problems.

Gradient-based optimization with categorical variables typically relies on score-function estimators, which are unbiased but noisy, or on continuous relaxations that replace the discrete distribution with a smooth surrogate admitting a pathwise (reparameterized) gradient, at the cost of optimizing a biased, temperature-dependent objective. In this paper, we extend this family of relaxations by introducing a diffusion-based soft reparameterization for categorical distributions. For these distributions, the denoiser under a Gaussian noising process admits a closed form and can be computed efficiently, yielding a training-free diffusion sampler through which we can backpropagate. Our experiments show that the proposed reparameterization trick yields competitive or improved optimization performance on various benchmarks.

Denoising and score estimation have long been known to be linked via the classical Tweedie's formula. In this work, we first extend the latter to a wider range of distributions often called "energy models" and denoted elliptical distributions in this work. Next, we examine an alternative view: we consider the denoising posterior $P(X|Y)$ as the optimizer of the energy score (a scoring rule) and derive a fundamental identity that connects the (path-) derivative of a (possibly) non-Euclidean energy score to the score of the noisy marginal. This identity can be seen as an analog of Tweedie's identity for the energy score, and allows for several interesting applications; for example, score estimation, noise distribution parameter estimation, as well as using energy score models in the context of "traditional" diffusion model samplers with a wider array of noising distributions.

In this work, we study the generalizability of diffusion models by looking into the hidden properties of the learned score functions, which are essentially a series of deep denoisers trained on various noise levels. We observe that as diffusion models transition from memorization to generalization, their corresponding nonlinear diffusion denoisers exhibit increasing linearity. This discovery leads us to investigate the linear counterparts of the nonlinear diffusion models, which are a series of linear models trained to match the function mappings of the nonlinear diffusion denoisers. Surprisingly, these linear denoisers are approximately the optimal denoisers for a multivariate Gaussian distribution characterized by the empirical mean and covariance of the training dataset. This finding implies that diffusion models have the inductive bias towards capturing and utilizing the Gaussian structure (covariance information) of the training dataset for data generation. We empirically demonstrate that this inductive bias is a unique property of diffusion models in the generalization regime, which becomes increasingly evident when the model's capacity is relatively small compared to the training dataset size. In the case that the model is highly overparameterized, this inductive bias emerges during the initial training phases before the model fully memorizes its training data. Our study provides crucial insights into understanding the notable strong generalization phenomenon recently observed in real-world diffusion models.

Recently developed deep-learning-based denoisers often outperform state-of-the-art conventional denoisers, such as the BM3D. They are typically trained to minimizethe mean squared error (MSE) between the output image of a deep neural networkand a ground truth image. In deep learning based denoisers, it is important to use high quality noiseless ground truth data for high performance, but it is often challenging or even infeasible to obtain noiseless images in application areas such as hyperspectral remote sensing and medical imaging. In this article, we propose a method based on Stein's unbiased risk estimator (SURE) for training deep neural network denoisers only based on the use of noisy images. We demonstrate that our SURE-based method, without the use of ground truth data, is able to train deep neural network denoisers to yield performances close to those networks trained with ground truth, and to outperform the state-of-the-art denoiser BM3D. Further improvements were achieved when noisy test images were used for training of denoiser networks using our proposed SURE-based method.