Neural network (NN) denoisers are an essential building block in many common
tasks, ranging from image reconstruction to image generation. However, the
success of these models is not well understood from a theoretical perspective.
In this paper, we aim to characterize the functions realized by shallow ReLU NN
denoisers -- in the common theoretical setting of interpolation (i.e., zero
training loss) with a minimal representation cost (i.e., minimal $\ell^2$ norm
weights). First, for univariate data, we derive a closed form for the NN
denoiser function, find it is contractive toward the clean data points, and
prove it generalizes better than the empirical MMSE estimator at a low noise
level. Next, for multivariate data, we find the NN denoiser functions in a
closed form under various geometric assumptions on the training data: data
contained in a low-dimensional subspace, data contained in a union of one-sided
rays, or several types of simplexes. These functions decompose into a sum of
simple rank-one piecewise linear interpolations aligned with edges and/or faces
connecting training samples. We empirically verify this alignment phenomenon on
synthetic data and real images.

深度学习去噪网络的理论性质研究，发现其函数具有包含简单线性插值的特征，并且通过实验证明其具有低噪声水平下超越经验 MMSE 估计器的泛化能力。

最小范数浅层降噪器在函数空间中的外貌如何？

How do Minimum-Norm Shallow Denoisers Look in Function Space?

We study the convergence of $Q$-learning with linear function approximation.
Our key contribution is the introduction of a novel multi-Bellman operator that
extends the traditional Bellman operator. By exploring the properties of this
operator, we identify conditions under which the projected multi-Bellman
operator becomes contractive, providing improved fixed-point guarantees
compared to the Bellman operator. To leverage these insights, we propose the
multi $Q$-learning algorithm with linear function approximation. We demonstrate
that this algorithm converges to the fixed-point of the projected multi-Bellman
operator, yielding solutions of arbitrary accuracy. Finally, we validate our
approach by applying it to well-known environments, showcasing the
effectiveness and applicability of our findings.

我们研究了具有线性函数逼近的 $Q$- 学习的收敛性。我们的关键贡献是引入了一个新颖的多 Bellman 算子，它扩展了传统的 Bellman 算子。通过探索该算子的属性，我们确定了投影的多 Bellman 算子变得收敛的条件，与 Bellman 算子相比，提供了改进的定点保证。为了利用这些认识，我们提出了具有线性函数逼近的多 $Q$- 学习算法。我们证明了该算法收敛到投影的多 Bellman 算子的稳定点，能够得到任意精度的解。最后，我们通过将其应用于众所周知的环境来验证我们的方法，展示了我们研究结果的有效性和适用性。