Kernels on graphs have had limited options for node-level problems. To address this, we present a novel, generalized kernel for graphs with node feature data for semi-supervised learning. The kernel is derived from a regularization framework by treating the graph and feature data as two Hilbert spaces. We also show how numerous kernel-based models on graphs are instances of our design. A kernel defined this way has transductive properties, and this leads to improved ability to learn on fewer training points, as well as better handling of highly non-Euclidean data. We demonstrate these advantages using synthetic data where the distribution of the whole graph can inform the pattern of the labels. Finally, by utilizing a flexible polynomial of the graph Laplacian within the kernel, the model also performed effectively in semi-supervised classification on graphs of various levels of homophily.

高斯过程（GPS）提供了对图表的推理和学习的原则和直接的方法。然而，缺乏用于时空建模的正义的图形内核已经备份了在图形问题中的使用。我们在图形上利用随机偏微分方程（SPDES）和GPS之间的显式链接，并导出捕获空间和时间交互的不可分离的时空图形内核。我们制定了随机热方程和波动方程的图形核。我们展示通过为图形提供新颖的时空GP建模的新型工具，我们在特征扩散，振荡和其他复杂交互中的实际应用中优先于现有的图形内核。

We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semi-supervised framework that incorporates labeled and unlabeled data in a general-purpose learner. Some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases. We use properties of reproducing kernel Hilbert spaces to prove new Representer theorems that provide theoretical basis for the algorithms. As a result (in contrast to purely graph-based approaches) we obtain a natural out-of-sample extension to novel examples and so are able to handle both transductive and truly semi-supervised settings. We present experimental evidence suggesting that our semi-supervised algorithms are able to use unlabeled data effectively. Finally we have a brief discussion of unsupervised and fully supervised learning within our general framework.

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。

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.

Data-driven neighborhood definitions and graph constructions are often used in machine learning and signal processing applications. k-nearest neighbor~(kNN) and $\epsilon$-neighborhood methods are among the most common methods used for neighborhood selection, due to their computational simplicity. However, the choice of parameters associated with these methods, such as k and $\epsilon$, is still ad hoc. We make two main contributions in this paper. First, we present an alternative view of neighborhood selection, where we show that neighborhood construction is equivalent to a sparse signal approximation problem. Second, we propose an algorithm, non-negative kernel regression~(NNK), for obtaining neighborhoods that lead to better sparse representation. NNK draws similarities to the orthogonal matching pursuit approach to signal representation and possesses desirable geometric and theoretical properties. Experiments demonstrate (i) the robustness of the NNK algorithm for neighborhood and graph construction, (ii) its ability to adapt the number of neighbors to the data properties, and (iii) its superior performance in local neighborhood and graph-based machine learning tasks.

这篇综述的目的是将读者介绍到图表内，以将其应用于化学信息学中的分类问题。图内核是使我们能够推断分子的化学特性的功能，可以帮助您完成诸如寻找适合药物设计的化合物等任务。内核方法的使用只是一种特殊的两种方式量化了图之间的相似性。我们将讨论限制在这种方法上，尽管近年来已经出现了流行的替代方法，但最著名的是图形神经网络。

神经切线核是根据无限宽度神经网络的参数分布定义的内核函数。尽管该极限不切实际，但神经切线内核允许对神经网络进行更直接的研究，并凝视着黑匣子的面纱。最近，从理论上讲，Laplace内核和神经切线内核在$ \ Mathbb {S}}^{D-1} $中共享相同的复制核Hilbert空间，暗示了它们的等价。在这项工作中，我们分析了两个内核的实际等效性。我们首先是通过与核的准确匹配，然后通过与高斯过程的后代匹配来进行匹配。此外，我们分析了$ \ mathbb {r}^d $中的内核，并在回归任务中进行实验。

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.

由于数据的注释可以在大规模的实际问题中稀缺，利用未标记的示例是机器学习中最重要的方面之一。这是半监督学习的目的。从访问未标记数据的访问中受益，它很自然地弥漫将标记数据平稳地知识到未标记的数据。这诱导了Laplacian正规化的使用。然而，Laplacian正则化的当前实施遭受了几种缺点，特别是众所周知的维度诅咒。在本文中，我们提供了统计分析以克服这些问题，并揭示了具有所需行为的大型光谱滤波方法。它们通过（再现）内核方法来实现，我们提供了现实的计算指南，以使我们的方法可用于大量数据。

In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogues to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting, and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.

最新提出的基于变压器的图形模型的作品证明了香草变压器用于图形表示学习的不足。要了解这种不足，需要研究变压器的光谱分析是否会揭示其对其表现力的见解。类似的研究已经确定，图神经网络（GNN）的光谱分析为其表现力提供了额外的观点。在这项工作中，我们系统地研究并建立了变压器领域中的空间和光谱域之间的联系。我们进一步提供了理论分析，并证明了变压器中的空间注意机制无法有效捕获所需的频率响应，因此，固有地限制了其在光谱空间中的表现力。因此，我们提出了feta，该框架旨在在整个图形频谱（即图形的实际频率成分）上进行注意力类似于空间空间中的注意力。经验结果表明，FETA在标准基准的所有任务中为香草变压器提供均匀的性能增益，并且可以轻松地扩展到具有低通特性的基于GNN的模型（例如GAT）。

在过去十年中，图形内核引起了很多关注，并在结构化数据上发展成为一种快速发展的学习分支。在过去的20年中，该领域发生的相当大的研究活动导致开发数十个图形内核，每个图形内核都对焦于图形的特定结构性质。图形内核已成功地成功地在广泛的域中，从社交网络到生物信息学。本调查的目标是提供图形内核的文献的统一视图。特别是，我们概述了各种图形内核。此外，我们对公共数据集的几个内核进行了实验评估，并提供了比较研究。最后，我们讨论图形内核的关键应用，并概述了一些仍有待解决的挑战。

光谱图神经网络是基于图信号过滤器的一种图神经网络（GNN）。一些能够学习任意光谱过滤器的模型最近出现了。但是，很少有作品分析光谱GNN的表达能力。本文理论上研究了光谱GNNS的表现力。我们首先证明，即使没有非线性的光谱GNN也可以产生任意的图形信号，并给出了两个条件以达到普遍性。它们是：1）图Laplacian的多个特征值和2）节点特征中没有缺失的频率组件。我们还建立了光谱GNN的表达能力与图形同构（GI）测试之间的联系，后者通常用于表征空间GNNS的表达能力。此外，我们从优化的角度研究了具有相同表达能力的不同光谱GNN之间的经验性能差异，并激发了其重量函数对应于光谱中图信号密度的正交基础的使用。受分析的启发，我们提出了Jacobiconv，该雅各比基的正交性和灵活性使用了雅各比的基础，以适应广泛的重量功能。 Jacobiconv抛弃了非线性，同时在合成和现实世界数据集上都超过了所有基线。

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.

网络邻接矩阵的光谱嵌入通常产生大约围绕低维子纤维结构的节点表示。特别地，当从潜在位置模型产生图表时，期望隐藏的子结构出现。此外，网络内的社区存在可能在嵌入中生成特定的特定社区的子多种结构，但是在网络的大多数统计模型中，这不明确地解释。在本文中，提出了一类称为潜在结构块模型（LSBM）的模型来解决这种情况，允许在存在社区特定的一维歧管结构时允许图形聚类。 LSBMS专注于特定的潜伏空间模型，随机点产品图（RDPG），并为每个社区的潜在位置分配潜在的子多种。讨论了来自LSBMS引起的嵌入式的贝叶斯模型，并显示在模拟和现实世界网络数据上具有良好的性能。该模型能够正确地恢复生活在一维歧管中的底层社区，即使当底层曲线的参数形式未知，也可以在各种实际数据上实现显着的结果。

众所周知，即使通过核心点之间捕获数据点之间的相似性，也可以通过捕获相似性来提供准确的预测和不确定性估计，以提供准确的预测和不确定性估计。然而，传统的GP内核在捕获高维数据点之间的相似性时不是非常有效的。神经网络可用于学习在高维数据中编码复杂结构的良好表示，并且可以用作GP内核的输入。然而，神经网络的巨大数据要求使得这种方法在小数据设置中无效。为了解决代表学习和数据效率的冲突问题，我们建议通过使用概率神经网络来学习概率嵌入的深核。我们的方法将高维数据映射到低维子空间中的概率分布，然后计算这些分布之间的内核以捕获相似性。要启用端到端学习，我们可以推导出用于培训模型的功能梯度血清过程。各种数据集的实验表明，我们的方法在监督和半监督设置中占GP内核学习中的最先进。我们还将我们的方法扩展到其他小型数据范例，例如少量分类，在迷你想象网和小熊数据集上以前的方式胜过先前的方法。

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing, along with a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas. We then summarize recent advances in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning.

Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online. 1

We propose a principled way to define Gaussian process priors on various sets of unweighted graphs: directed or undirected, with or without loops. We endow each of these sets with a geometric structure, inducing the notions of closeness and symmetries, by turning them into a vertex set of an appropriate metagraph. Building on this, we describe the class of priors that respect this structure and are analogous to the Euclidean isotropic processes, like squared exponential or Mat\'ern. We propose an efficient computational technique for the ostensibly intractable problem of evaluating these priors' kernels, making such Gaussian processes usable within the usual toolboxes and downstream applications. We go further to consider sets of equivalence classes of unweighted graphs and define the appropriate versions of priors thereon. We prove a hardness result, showing that in this case, exact kernel computation cannot be performed efficiently. However, we propose a simple Monte Carlo approximation for handling moderately sized cases. Inspired by applications in chemistry, we illustrate the proposed techniques on a real molecular property prediction task in the small data regime.