The utilisation of Plug-and-Play (PnP) priors in inverse problems has become
increasingly prominent in recent years. This preference is based on the
mathematical equivalence between the general proximal operator and the
regularised denoiser, facilitating the adaptation of various off-the-shelf
denoiser priors to a wide range of inverse problems. However, existing PnP
models predominantly rely on pre-trained denoisers using large datasets. In
this work, we introduce Single-Shot PnP methods (SS-PnP), shifting the focus to
solving inverse problems with minimal data. First, we integrate Single-Shot
proximal denoisers into iterative methods, enabling training with single
instances. Second, we propose implicit neural priors based on a novel function
that preserves relevant frequencies to capture fine details while avoiding the
issue of vanishing gradients. We demonstrate, through extensive numerical and
visual experiments, that our method leads to better approximations.

利用 Plug-and-Play（PnP）先验在逆问题中的应用日益突出，本研究介绍了 Single-Shot PnP 方法（SS-PnP），使用少量数据解决逆问题，通过将 Single-Shot 近端去噪器集成到迭代方法中，并提出基于保留相关频率的新型函数的隐式神经先验，以获得更好的逼近。

单次插拔式逆问题解决方案

Single-Shot Plug-and-Play Methods for Inverse Problems

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP)
methods have become ubiquitous in Bayesian imaging. These methods derive
Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for
inverse problems in imaging by combining an explicit likelihood function with a
prior that is implicitly defined by an image denoising algorithm. The PnP
algorithms proposed in the literature mainly differ in the iterative schemes
they use for optimisation or for sampling. In the case of optimisation schemes,
some recent works guarantee the convergence to a fixed point, albeit not
necessarily a MAP estimate. In the case of sampling schemes, to the best of our
knowledge, there is no known proof of convergence. There also remain important
open questions regarding whether the underlying Bayesian models and estimators
are well defined, well-posed, and have the basic regularity properties required
to support these numerical schemes. To address these limitations, this paper
develops theory, methods, and provably convergent algorithms for performing
Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA
(Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference;
and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent
results on the quantitative convergence of Markov chains, we establish detailed
convergence guarantees for these two algorithms under realistic assumptions on
the denoising operators used, with special attention to denoisers based on deep
neural networks. We also show that these algorithms approximately target a
decision-theoretically optimal Bayesian model that is well-posed. The proposed
algorithms are demonstrated on several canonical problems such as image
deblurring, inpainting, and denoising, where they are used for point estimation
as well as for uncertainty visualisation and quantification.

本文介绍了两种算法：PnP-ULA（未调整的 Langevin 算法）和 PnP-SGD（随机梯度下降）, 用于进行具有 Plug & Play 先验的贝叶斯推断，其中深度神经网络去噪算子特别值得注意，同时针对图像去模糊、修复和去噪等标准问题，这些算法可用于点估计和不确定性可视化和量化。