散射变换是一种基于多层的小波的深度学习架构，其充当卷积神经网络的模型。最近，几种作品引入了非欧几里德设置的散射变换的概括，例如图形。我们的工作通过基于非常一般的非对称小波来引入图形的窗口和非窗口几何散射变换来构建这些结构。我们表明，这些不对称的图形散射变换具有许多与其对称对应的相同的理论保证。结果，所提出的结构统一并扩展了许多现有图散射架构的已知理论结果。在这样做时，这项工作有助于通过引入具有可提供稳定性和不变性保证的大型网络，帮助弥合几何散射和其他图形神经网络之间的差距。这些结果为未来的图形结构数据奠定了基础，对具有学习过滤器的图形结构数据，并且还可以证明具有理想的理论特性。

散射变换是一种基于小波的多层转换，最初是作为卷积神经网络（CNN）的模型引入的，它在我们对这些网络稳定性和不变性属性的理解中发挥了基础作用。随后，人们普遍兴趣将CNN的成功扩展到具有非欧盟结构的数据集，例如图形和歧管，从而导致了几何深度学习的新兴领域。为了提高我们对这个新领域中使用的体系结构的理解，几篇论文提出了对非欧几里得数据结构（如无方向的图形和紧凑的Riemannian歧管）的散射转换的概括。在本文中，我们介绍了一个通用的统一模型，用于测量空间上的几何散射。我们提出的框架包括以前的几何散射作品作为特殊情况，但也适用于更通用的设置，例如有向图，签名图和带边界的歧管。我们提出了一个新标准，该标准可以识别哪些有用表示应该不变的组，并表明该标准足以确保散射变换具有理想的稳定性和不变性属性。此外，我们考虑从随机采样未知歧管获得的有限度量空间。我们提出了两种构造数据驱动图的方法，在该图上相关的图形散射转换近似于基础歧管上的散射变换。此外，我们使用基于扩散图的方法来证明这些近似值之一的收敛速率的定量估计值，因为样品点的数量趋向于无穷大。最后，我们在球形图像，有向图和高维单细胞数据上展示了方法的实用性。

几何深度学习取得了长足的进步，旨在概括从传统领域到非欧几里得群岛的结构感知神经网络的设计，从而引起图形神经网络（GNN），这些神经网络（GNN）可以应用于形成的图形结构数据，例如社会，例如，网络，生物化学和材料科学。尤其是受欧几里得对应物的启发，尤其是图形卷积网络（GCN）通过提取结构感知功能来成功处理图形数据。但是，当前的GNN模型通常受到各种现象的限制，这些现象限制了其表达能力和推广到更复杂的图形数据集的能力。大多数模型基本上依赖于通过本地平均操作对图形信号的低通滤波，从而导致过度平滑。此外，为了避免严重的过度厚度，大多数流行的GCN式网络往往是较浅的，并且具有狭窄的接收场，导致侵犯。在这里，我们提出了一个混合GNN框架，该框架将传统的GCN过滤器与通过几何散射定义的带通滤波器相结合。我们进一步介绍了一个注意框架，该框架允许该模型在节点级别上从不同过滤器的组合信息进行本地参与。我们的理论结果确定了散射过滤器的互补益处，以利用图表中的结构信息，而我们的实验显示了我们方法对各种学习任务的好处。

我们研究光谱图卷积神经网络（GCNN），其中过滤器被定义为通过功能计算的图形移位算子（GSO）的连续函数。光谱GCNN不是针对一个特定图的量身定制的，可以在不同的图之间传输。因此，研究GCNN的可传递性很重要：网络在代表相同现象的不同图上具有大致相同影响的能力。如果测试集中的图与训练集中的图形相同，则可传递性可确保在某些图上进行训练的GCNN概括。在本文中，我们考虑了基于Graphon分析的可转让性模型。图形是图形的极限对象，在图形范式中，如果两者都近似相同的图形，则两个图表示相同的现象。我们的主要贡献可以总结如下：1）我们证明，在近似于同一图形的图的图下，任何具有连续过滤器的固定GCNN都是可以转移的，2）我们证明了近似于未结合的图形换档运算符的图形，该图是在本文中定义的，和3）我们获得了非反应近似结果，证明了GCNN的线性稳定性。这扩展了当前的最新结果，这些结果显示了在近似界图子的图下显示多项式过滤器的渐近可传递性。

We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T t g = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.

图形神经网络（GNNS）是由图形卷积和叉指非线性组成的层组成的深度卷积架构。由于其不变性和稳定性属性，GNN在网络数据的学习陈述中被证明是成功的。但是，训练它们需要矩阵计算，这对于大图可能是昂贵的。为了解决这个限制，我们研究了GNN横跨图形转移的能力。我们考虑图形，这是加权和随机图形的图形限制和生成模型，以定义图形卷积和GNNS - Graphon卷曲和Graphon神经网络（WNNS）的限制对象 - 我们用作图形卷曲的生成模型和GNNS。我们表明，这些石墨源区和WNN可以通过图形滤波器和来自加权和随机图中的它们采样的GNN来近似。使用这些结果，我们将导出误差界限，用于跨越此类图形传输图形过滤器和GNN。这些界限表明，可转换性随着图尺寸的增加而增加，并且揭示了在GNN中的可转换性和光谱分辨率之间的折衷，其被点亮的非线性缓解。这些发现经验在电影推荐和分散机器人控制中的数值实验中进行了经验验证。

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them.Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field.

图表表示学习有许多现实世界应用，从超级分辨率的成像，3D计算机视觉到药物重新扫描，蛋白质分类，社会网络分析。图表数据的足够表示对于图形结构数据的统计或机器学习模型的学习性能至关重要。在本文中，我们提出了一种用于图形数据的新型多尺度表示系统，称为抽取帧的图形数据，其在图表上形成了本地化的紧密框架。抽取的帧系统允许在粗粒链上存储图形数据表示，并在每个比例的多个尺度处处理图形数据，数据存储在子图中。基于此，我们通过建设性数据驱动滤波器组建立用于在多分辨率下分解和重建图数据的抽取G-Framewelet变换。图形帧构建基于基于链的正交基础，支持快速图傅里叶变换。由此，我们为抽取的G-Frameword变换或FGT提供了一种快速算法，该算法具有线性计算复杂度O（n），用于尺寸N的图表。用数值示例验证抽取的帧谱和FGT的理论，用于随机图形。现实世界应用的效果是展示的，包括用于交通网络的多分辨率分析，以及图形分类任务的图形神经网络。

我们提出了一个新的图神经网络（GNN）模块，该模块基于最近提出的几何散射变换的松弛，该变换由图形小波滤波器组成。我们可学习的几何散射（腿）模块可以使小波的自适应调整能够鼓励乐队通道特征在学习的表示中出现。与许多流行的GNN相比，我们的腿部模块在GNN中的结合能够学习长期图形关系，这些GNN通常依赖于邻居之间的平滑度或相似性来编码图形结构。此外，与竞争性GNN相比，其小波先验会导致简化的架构，学到的参数明显少得多。我们证明了基于腿的网络在图形分类基准上的预测性能，以及在生化图数据探索任务中学到的功能的描述性质量。我们的结果表明，基于腿部的网络匹配或匹配流行的GNN，以及在许多数据集上，尤其是在生化域中的原始几何散射结构，同时保留了手工制作的（非学习）几何散射的某些数学特性。

Network data are ubiquitous in modern machine learning, with tasks of interest including node classification, node clustering and link prediction. A frequent approach begins by learning an Euclidean embedding of the network, to which algorithms developed for vector-valued data are applied. For large networks, embeddings are learned using stochastic gradient methods where the sub-sampling scheme can be freely chosen. Despite the strong empirical performance of such methods, they are not well understood theoretically. Our work encapsulates representation methods using a subsampling approach, such as node2vec, into a single unifying framework. We prove, under the assumption that the graph is exchangeable, that the distribution of the learned embedding vectors asymptotically decouples. Moreover, we characterize the asymptotic distribution and provided rates of convergence, in terms of the latent parameters, which includes the choice of loss function and the embedding dimension. This provides a theoretical foundation to understand what the embedding vectors represent and how well these methods perform on downstream tasks. Notably, we observe that typically used loss functions may lead to shortcomings, such as a lack of Fisher consistency.

许多监督的学习问题涉及高维数据，例如图像，文本或图形。为了能够有效地利用数据，它通常有用的是在手头的问题中利用某些几何前瞻，例如与换算，置换子组或稳定性的不变性。通过考虑球体上这些功能的球形谐波分解，我们研究了目标功能提出了这种不变性和稳定性特性的学习问题的样本复杂性。我们提供内核方法的非参数率的收敛速度，并且在与相应的非不变内核相比，在该组上使用不变内核时，通过等于组的大小的因子的提高。当样本大小足够大时，这些改进是有效的，其渐近行为取决于该组的光谱特性。最后，这些增益扩展到不变性组之外，还涵盖小变形的几何稳定性，这里被建模为排列的子集（不一定是子组）。

Pre-publication draft of a book to be published byMorgan & Claypool publishers. Unedited version released with permission. All relevant copyrights held by the author and publisher extend to this pre-publication draft.

High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a method for constructing manifold neural networks using the spectral decomposition of the Laplace Beltrami operator. Moreover, in this work, the authors provide a numerical scheme for implementing such neural networks when the manifold is unknown and one only has access to finitely many sample points. The authors show that this scheme, which relies upon building a data-driven graph, converges to the continuum limit as the number of sample points tends to infinity. Here, we build upon this result by establishing a rate of convergence that depends on the intrinsic dimension of the manifold but is independent of the ambient dimension. We also discuss how the rate of convergence depends on the depth of the network and the number of filters used in each layer.

Graph Neural Networks (graph NNs) are a promising deep learning approach for analyzing graph-structured data. However, it is known that they do not improve (or sometimes worsen) their predictive performance as we pile up many layers and add non-lineality. To tackle this problem, we investigate the expressive power of graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our strategy is to generalize the forward propagation of a Graph Convolutional Network (GCN), which is a popular graph NN variant, as a specific dynamical system. In the case of a GCN, we show that when its weights satisfy the conditions determined by the spectra of the (augmented) normalized Laplacian, its output exponentially approaches the set of signals that carry information of the connected components and node degrees only for distinguishing nodes. Our theory enables us to relate the expressive power of GCNs with the topological information of the underlying graphs inherent in the graph spectra. To demonstrate this, we characterize the asymptotic behavior of GCNs on the Erdős -Rényi graph. We show that when the Erdős -Rényi graph is sufficiently dense and large, a broad range of GCNs on it suffers from the "information loss" in the limit of infinite layers with high probability. Based on the theory, we provide a principled guideline for weight normalization of graph NNs. We experimentally confirm that the proposed weight scaling enhances the predictive performance of GCNs in real data 1 .

Although theoretical properties such as expressive power and over-smoothing of graph neural networks (GNN) have been extensively studied recently, its convergence property is a relatively new direction. In this paper, we investigate the convergence of one powerful GNN, Invariant Graph Network (IGN) over graphs sampled from graphons. We first prove the stability of linear layers for general $k$-IGN (of order $k$) based on a novel interpretation of linear equivariant layers. Building upon this result, we prove the convergence of $k$-IGN under the model of \citet{ruiz2020graphon}, where we access the edge weight but the convergence error is measured for graphon inputs. Under the more natural (and more challenging) setting of \citet{keriven2020convergence} where one can only access 0-1 adjacency matrix sampled according to edge probability, we first show a negative result that the convergence of any IGN is not possible. We then obtain the convergence of a subset of IGNs, denoted as IGN-small, after the edge probability estimation. We show that IGN-small still contains function class rich enough that can approximate spectral GNNs arbitrarily well. Lastly, we perform experiments on various graphon models to verify our statements.

我们介绍了一种新颖的谐波分析，用于在函数上定义的函数，随机步行操作员是基石。作为第一步，我们将随机步行操作员的一组特征向量作为非正交傅里叶类型的功能，用于通过定向图。我们通过将从其Dirichlet能量获得的随机步行操作员的特征向量的变化与其相关的特征值的真实部分连接来发现频率解释。从这个傅立叶基础，我们可以进一步继续，并在有向图中建立多尺度分析。通过将Coifman和MagGioni扩展到定向图，我们提出了一种冗余小波变换和抽取的小波变换。因此，我们对导向图的谐波分析的发展导致我们考虑应用于突出了我们框架效率的指示图的图形上的半监督学习问题和信号建模问题。

我们为特殊神经网络架构，称为运营商复发性神经网络的理论分析，用于近似非线性函数，其输入是线性运算符。这些功能通常在解决方案算法中出现用于逆边值问题的问题。传统的神经网络将输入数据视为向量，因此它们没有有效地捕获与对应于这种逆问题中的数据的线性运算符相关联的乘法结构。因此，我们介绍一个类似标准的神经网络架构的新系列，但是输入数据在向量上乘法作用。由较小的算子出现在边界控制中的紧凑型操作员和波动方程的反边值问题分析，我们在网络中的选择权重矩阵中促进结构和稀疏性。在描述此架构后，我们研究其表示属性以及其近似属性。我们还表明，可以引入明确的正则化，其可以从所述逆问题的数学分析导出，并导致概括属性上的某些保证。我们观察到重量矩阵的稀疏性改善了概括估计。最后，我们讨论如何将运营商复发网络视为深度学习模拟，以确定诸如用于从边界测量的声波方程中重建所未知的WAVESTED的边界控制的算法算法。

在本文中，我们为基于非交换代数的代数神经网络（ALGNN）提供稳定性结果。 ALGNN是堆叠的分层结构，每个层都与代数信号模型（ASM）相关联，由代数，矢量空间和同态性。信号被建模为矢量空间的元素，过滤器是代数中的元素，而同态则可以实现过滤器作为混凝土操作员。我们研究了代数过滤器在非交换代数对同态扰动中的稳定性，并提供了保证稳定性的条件。我们表明，轮班运算符和偏移和扰动之间的换向性不会影响稳定体系结构的属性。这提供了一个问题，即转移不变性是否是保证稳定性的卷积体系结构的必要属性。此外，我们表明，尽管非交换代数中过滤器的频率响应在交换代数中与过滤器相对于过滤器表现出很大的差异，但它们的稳定过滤器的衍生物具有相似的行为。

We consider the problem of estimating a multivariate function $f_0$ of bounded variation (BV), from noisy observations $y_i = f_0(x_i) + z_i$ made at random design points $x_i \in \mathbb{R}^d$, $i=1,\ldots,n$. We study an estimator that forms the Voronoi diagram of the design points, and then solves an optimization problem that regularizes according to a certain discrete notion of total variation (TV): the sum of weighted absolute differences of parameters $\theta_i,\theta_j$ (which estimate the function values $f_0(x_i),f_0(x_j)$) at all neighboring cells $i,j$ in the Voronoi diagram. This is seen to be equivalent to a variational optimization problem that regularizes according to the usual continuum (measure-theoretic) notion of TV, once we restrict the domain to functions that are piecewise constant over the Voronoi diagram. The regression estimator under consideration hence performs (shrunken) local averaging over adaptively formed unions of Voronoi cells, and we refer to it as the Voronoigram, following the ideas in Koenker (2005), and drawing inspiration from Tukey's regressogram (Tukey, 1961). Our contributions in this paper span both the conceptual and theoretical frontiers: we discuss some of the unique properties of the Voronoigram in comparison to TV-regularized estimators that use other graph-based discretizations; we derive the asymptotic limit of the Voronoi TV functional; and we prove that the Voronoigram is minimax rate optimal (up to log factors) for estimating BV functions that are essentially bounded.

We introduce an architecture for processing signals supported on hypergraphs via graph neural networks (GNNs), which we call a Hyper-graph Expansion Neural Network (HENN), and provide the first bounds on the stability and transferability error of a hypergraph signal processing model. To do so, we provide a framework for bounding the stability and transferability error of GNNs across arbitrary graphs via spectral similarity. By bounding the difference between two graph shift operators (GSOs) in the positive semi-definite sense via their eigenvalue spectrum, we show that this error depends only on the properties of the GNN and the magnitude of spectral similarity of the GSOs. Moreover, we show that existing transferability results that assume the graphs are small perturbations of one another, or that the graphs are random and drawn from the same distribution or sampled from the same graphon can be recovered using our approach. Thus, both GNNs and our HENNs (trained using normalized Laplacians as graph shift operators) will be increasingly stable and transferable as the graphs become larger. Experimental results illustrate the importance of considering multiple graph representations in HENN, and show its superior performance when transferability is desired.