Gaussian denoising has emerged as a powerful principle for constructing
simulation-free continuous normalizing flows for generative modeling. Despite
their empirical successes, theoretical properties of these flows and the
regularizing effect of Gaussian denoising have remained largely unexplored. In
this work, we aim to address this gap by investigating the well-posedness of
simulation-free continuous normalizing flows built on Gaussian denoising.
Through a unified framework termed Gaussian interpolation flow, we establish
the Lipschitz regularity of the flow velocity field, the existence and
uniqueness of the flow, and the Lipschitz continuity of the flow map and the
time-reversed flow map for several rich classes of target distributions. This
analysis also sheds light on the auto-encoding and cycle-consistency properties
of Gaussian interpolation flows. Additionally, we delve into the stability of
these flows in source distributions and perturbations of the velocity field,
using the quadratic Wasserstein distance as a metric. Our findings offer
valuable insights into the learning techniques employed in Gaussian
interpolation flows for generative modeling, providing a solid theoretical
foundation for end-to-end error analyses of learning GIFs with empirical
observations.

我们研究了基于高斯降噪的模拟 - free 连续标准化流的良定义性、可微性和唯一性，通过建立高斯插值流的统一框架，我们证明了流速度场的利普希茨正则性、流的存在唯一性，以及流映射和时间反向流映射在多个丰富的目标分布类别中的利普希茨连续性，这也揭示了高斯插值流的自动编码和循环一致性特性。我们还研究了这些流在源分布和速度场扰动中的稳定性，利用二次 Wasserstein 距离作为度量。我们的发现为用于生成建模中的高斯插值流的学习技术提供了宝贵的见解，为使用经验观察进行端到端误差分析的学习 GIFs 提供了坚实的理论基础。

高斯插值流

Gaussian Interpolation Flows

A common approach to solve inverse imaging problems relies on finding a
maximum a posteriori (MAP) estimate of the original unknown image, by solving a
minimization problem. In thiscontext, iterative proximal algorithms are widely
used, enabling to handle non-smooth functions and linear operators. Recently,
these algorithms have been paired with deep learning strategies, to further
improve the estimate quality. In particular, proximal neural networks (PNNs)
have been introduced, obtained by unrolling a proximal algorithm as for finding
a MAP estimate, but over a fixed number of iterations, with learned linear
operators and parameters. As PNNs are based on optimization theory, they are
very flexible, and can be adapted to any image restoration task, as soon as a
proximal algorithm can solve it. They further have much lighter architectures
than traditional networks. In this article we propose a unified framework to
build PNNs for the Gaussian denoising task, based on both the dual-FB and the
primal-dual Chambolle-Pock algorithms. We further show that accelerated
inertial versions of these algorithms enable skip connections in the associated
NN layers. We propose different learning strategies for our PNN framework, and
investigate their robustness (Lipschitz property) and denoising efficiency.
Finally, we assess the robustness of our PNNs when plugged in a
forward-backward algorithm for an image deblurring problem.

在这篇文章中，我们提出了一个统一的框架来构建用于高斯降噪任务的近端神经网络（PNNs），该框架基于双重 - 投影算法和原 - 对偶 Chambolle-Pock 算法。我们还展示了这些算法的加速惯性版本如何在关联的神经网络层中实现跳跃连接。我们为 PNN 框架提供了不同的学习策略，并研究了它们的鲁棒性（利普希茨性质）和降噪效率。最后，我们评估了将我们的 PNNs 插入图像去模糊问题的前向 - 后向算法时的鲁棒性。

PNN：从近端算法到鲁棒展开图像去噪网络和即插即用方法

PNN: From proximal algorithms to robust unfolded image denoising  networks and Plug-and-Play methods

In recent years, Deep Convolutional Neural Networks (DCNNs) have outreached
the performance of classical algorithms for image restoration tasks. However
most of these methods are not suited for computational efficiency and are
therefore too expensive to be executed on embedded and mobile devices. In this
work we investigate Spiking Neural Networks (SNNs) for Gaussian denoising, with
the goal of approaching the performance of conventional DCNN while reducing the
computational load. We propose a formal analysis of the information conversion
processing carried out by the Leaky Integrate and Fire (LIF) neurons and we
compare its performance with the classical rate-coding mechanism. The neural
coding schemes are then evaluated through experiments in terms of denoising
performance and computation efficiency for a state-of-the-art deep
convolutional neural network. Our results show that SNNs with LIF neurons can
provide competitive denoising performance but at a reduced computational cost.

通过使用 Leaky Integrate and Fire 神经元，我们研究了脉冲神经网络在高斯去噪任务中的性能，结果表明相对于深度卷积神经网络，SNN 能够在降低计算负载的情况下提供有竞争力的去噪性能。