度量学习 流形学习_流形学习2

潜图深度学习 (Deep learning with latent graphs)

TL;DR: Graph neural networks exploit relational inductive biases for data that come in the form of a graph. However, in many cases we do not have the graph readily available. Can graph deep learning still be applied in this case? In this post, I draw parallels between recent works on latent graph learning and older techniques of manifold learning.

TL; DR： 图神经网络利用关系归纳偏差来处理图形式的数据。 但是，在许多情况下，我们没有图可用。 在这种情况下，仍然可以应用图深度学习吗？ 在这篇文章中，我将有关潜图学习的最新工作与较早的流形学习技术进行了比较。

The past few years have witnessed a surge of interest in developing ML methods for graph-structured data. Such data naturally arises in many applications such as social sciences (e.g. the Follow graph of users on Twitter or Facebook), chemistry (where molecules can be modelled as graphs of atoms connected by bonds), or biology (where interactions between different biomolecules are often modelled as a graph referred to as the interactome). Graph neural networks (GNNs), which I have covered extensively in my previous posts, are a particularly popular method of learning on such graphs by means of local operations with shared parameters exchanging information between adjacent nodes.

在过去的几年中，目睹了针对图结构数据开发ML方法的兴趣激增。 这样的数据自然会出现在许多应用中，例如社会科学(例如Twitter或Facebook上的“关注用户”图)，化学(其中的分子可以建模为通过键连接的原子的图)或生物学(其中不同生物分子之间经常发生相互作用)建模为称为交互组的图表)。 图神经网络(GNN)是我在以前的文章中广泛讨论的一种特别流行的方法，它是通过局部操作和共享参数在相邻节点之间交换信息来学习此类图。

In some settings, however, we do not have the luxury of being given a graph to work with as input. This is a typical situation for many biological problems, where graphs such as protein-to-protein interaction are only partially known in the best case, as the experiments by which interactions are discovered are expensive and noisy. We are therefore interested in inferring the graph from the data and applying a GNN on it [1] — I call this setting “latent graph learning” [2]. The latent graph might be application-specific and optimised for the downstream task. Furthermore, sometimes such a graph might be even more important than the task itself, as it may convey important insights about the data and offer a way to interpret the results.

但是，在某些情况下，我们不能奢侈地得到一个可作为输入使用的图形。 这是许多生物学问题的典型情况，在最好的情况下，仅部分了解蛋白质-蛋白质相互作用等图，因为发现相互作用的实验既昂贵又嘈杂。 因此，我们有兴趣根据数据推断图形并对其应用GNN [1]-我将此设置称为“潜在图形学习” [2]。 潜在图可能是特定于应用程序的，并针对下游任务进行了优化。 此外，有时这样的图形可能比任务本身更为重要，因为它可以传达有关数据的重要见解并提供一种解释结果的方法。

A way of thinking of latent graph learning is that of a graph with an empty edge set. In this setting, the input is provided as a point cloud in some high-dimensional feature space. Unlike methods for deep learning on sets such as PointNet [3], which apply some shared learnable point-wise function to each point, we also seek to pass information across points. This is done by message passing on a graph constructed from the point features themselves.

潜伏图学习的思维方式是一个空边集图的。 在这种设置下，输入被提供为某些高维特征空间中的点云。 与诸如PointNet [3]之类的在集合上进行深度学习的方法不同，该方法将共享的可学习的逐点函数应用于每个点，我们还寻求跨点传递信息。 这是通过将消息传递到由点要素本身构造的图形上来完成的。

The first architecture of this kind, Dynamic Graph CNN (DGCNN) [4], was developed by Yue Wang from MIT, with whom I had the pleasure to collaborate during my sabbatical at that institution. Motivated by problems in computer graphics dealing with the analysis of 3D point clouds, the idea was to use the graph as a coarse representation of the local smooth manifold structure underlying a point cloud. A key observation of Yue was that the graph does not need to stay the same throughout the neural network, and in fact it can and should be updated dynamically — hence the name of the method. The following figure from our paper illustrates why this might be useful in computer graphics problems:

麻省理工学院的王悦开发了这种第一种架构，即动态图CNN(DGCNN)[4]，在我休假期间，我很高兴与他合作。 受计算机图形学中涉及3D点云分析的问题的启发，其想法是将图形用作点云下局部光滑流形结构的粗略表示。 Yue的一个主要观察结果是，该图不需要在整个神经网络中保持不变，并且实际上并且可以并且应该动态地进行更新-因此该方法的名称。 我们论文的下图说明了为什么这可能在计算机图形问题中有用：

One of the limitations of DGCNNs was that the same space is used to construct the graph and the features on that graph. In a recent work with Anees Kazi from TUM and my postdoc Luca Cosmo, we proposed a new architecture called Differentiable Graph Module (DGM) [5] extending DGCNN by decoupling the graph and feature construction, as shown in the following figure:

DGCNN的局限性之一是使用相同的空间来构造图和该图上的特征。 在TUM的Anees Kazi和我的博士后Luca Cosmo的近期工作中，我们提出了一种新的架构，称为可微图模块(DGM)[5]，它通过将图和特征构造分离来扩展DGCNN，如下图所示：

DGM showed impressive results when applied to problems from the medical domain, such as disease prediction from brain imaging data. In these tasks, we are provided with the electronic health records of multiple patients, including demographic features (such as age, sex, etc) and brain imaging features, and attempt to predict whether a patient suffers from a neurological disease. Previous works showed the application of GNNs to such tasks using diffusion on a “patient graph” constructed by hand from the demographic features [6]. DGM offers the advantage of learning the graph, which possibly conveys insight into how certain features depend on each other for the specific diagnosis task. As a bonus, DGM also beat DGCNN in its game of point cloud classification, albeit only slightly.

将DGM应用于医学领域的问题时，例如从大脑成像数据进行疾病预测，其结果令人印象深刻。 在这些任务中，我们将获得多名患者的电子健康记录，包括人口统计特征(例如年龄，性别等)和大脑成像特征，并尝试预测患者是否患有神经系统疾病。 先前的工作表明，通过在人口统计特征上手工构建的“患者图”上进行扩散，将GNN应用于此类任务[6]。 DGM提供了学习图形的优势，这可能传达了对于特定诊断任务某些功能如何相互依赖的见解。 作为奖励，DGM在点云分类游戏中也击败了DGCNN，尽管只是稍有下降。

DGCNN and DGM bear conceptual similarity to a family of algorithms called manifold learning or non-linear dimensionality reduction, which were extremely popular in machine learning when I was a student in the 2000s, and are still used for data visualisation. The assumption underlying manifold learning methods is that of the data having an intrinsic low-dimensional structure. Though the data can be represented in a space of hundreds or even thousands of dimensions, it only has a few degrees of freedom, as shown in the following example:

D GCNN和DGM与称为流形学习或非线性降维的一系列算法在概念上相似，这在2000年代我还是学生的时候在机器学习中非常流行，并且仍然用于数据可视化。 多种学习方法的基础假设是具有固有的低维结构的数据。 尽管可以在数百个甚至数千个维的空间中表示数据，但它只有几个自由度，如以下示例所示：

The purpose of manifold learning is to capture these degrees of freedom (by reconstructing the underlying “manifold”, hence the name [7]) and reduce the dimensionality of the data to its intrinsic dimension. The important difference from linear dimensionality reduction such as PCA is that, due to the non-Euclidean structure of the data, there might be no possibility to recover the manifold by means of a linear projection [8]:

多重学习的目的是捕获这些自由度(通过重构底层的“流形”，因此得名[7])并将数据的维数减少到其固有维数。 与线性降维(例如PCA )的重要区别在于，由于数据的非欧几里德结构，可能无法通过线性投影来恢复流形[8]：

Manifold learning algorithms vary in the way they approach the recovery of the “manifold”, but share a common blueprint. First, they create a representation of the data, which is typically done by constructing a k-nearest neighbour graph capturing its local structure. Second, they compute a low-dimensional representation (embedding) of the data trying to preserve the structure of the original data. This is where most manifold learning methods differ. For example, Isomap [9] tries to preserve the graph geodesic distance, Locally Linear Embedding [10] finds a local representation of adjacent points, and Laplacian eigenmaps [11] use the eigenfunctions of the graph Laplacian operator as the low-dimensional embedding. This new representation “flattens” the original non-Euclidean structure into a Euclidean space that is easier to deal with. Third, once the representation is computed, a machine learning algorithm (typically clustering) is applied to it.

中号 anifold学习算法，他们的做法“总管”的恢复方式有所不同，但都有一个共同的蓝图。 首先，它们创建数据的表示形式，通常是通过构造捕获其局部结构的k近邻图来完成的。 其次，它们计算数据的低维表示(嵌入)，以尝试保留原始数据的结构。 这是大多数多种学习方法不同的地方。 例如，Isomap [9]尝试保留图的测地距离，Locally Linear Embedding [10]查找相邻点的局部表示，而Laplacian特征图[11]使用图Laplacian算子的特征函数作为低维嵌入。 这种新的表示形式将原始的非欧几里得结构“平化”为一个更易于处理的欧几里得空间。 第三，一旦计算出表示，就对其应用机器学习算法(通常为聚类)。

One of the challenges is that the construction of the graph is decoupled from the ML algorithm, and sometimes delicate parameter tuning (e.g. the number of neighbours or the neighbourhood radius) is needed in order to figure out how to build the graph to make the downstream task work well. Perhaps a far more serious drawback of manifold learning algorithms is that data rarely presents itself as low-dimensional in its native form. When dealing with images, for example, various handcrafted feature extraction techniques had to be used as pre-processing steps.

挑战ØNE的是，图的构造从ML算法解耦，有时细腻的参数调整(例如邻居的数量或附近半径)是必要的，以找出如何构建图形，使下游任务运行良好。 流形学习算法的一个更为严重的缺点是，数据很少以其本机形式呈现为低维。 例如，在处理图像时，必须将各种手工特征提取技术用作预处理步骤。

Graph deep learning offers a modern take on this process, by replacing this three-stage process outlined above with a single graph neural network. In dynamic graph CNNs or DGM, for instance, the construction of the graph and the learning are part of the same architecture:

图深度学习通过用单个图神经网络代替上面概述的这个三阶段过程，为该过程提供了现代的方法。 例如，在动态图CNN或DGM中，图的构造和学习是同一体系结构的一部分：

The appeal of this approach is the possibility to combine the treatment of individual data points and the space in which they reside in the same pipeline. In the example of images, one could use traditional CNNs to extract the visual features from each image and use a GNN to model the relations between them. This approach was used in the work of my PhD student Jan Svoboda: he proposed a graph-based regularisation layer (called PeerNet) for CNNs that allows to exchange information between multiple images [12]. PeerNets bear similarity to non-local means filters [13] in the way they aggregate information from multiple locations, with the main difference that the aggregation happens across multiple images rather than a single one. We showed that such a regularisation dramatically reduces the effect of adversarial perturbations to which standard CNNs are highly susceptible [14].

这种方法的吸引力在于，可以将单个数据点的处理方式和它们在同一管道中所驻留的空间相结合。 在图像示例中，可以使用传统的CNN从每张图像中提取视觉特征，并使用GNN对它们之间的关系进行建模。 我的博士生Jan Svoboda的工作中使用了这种方法：他为CNN提出了一个基于图的正则化层(称为PeerNet)，该层允许在多个图像之间交换信息[12]。 PeerNet与非本地均值过滤器[13]的相似之处在于它们聚合来自多个位置的信息的方式，主要区别在于聚合发生在多个图像而不是单个图像上。 我们表明，这种正则化极大地降低了标准CNN极易受到对抗性干扰的影响[14]。

There are many other interesting applications of latent graph learning. One is few-shot learning, where graph-based techniques can help generalise from a few examples. Few-shot learning is becoming increasingly important in computer vision where the cost of data labelling is significant [5]. Another field is biology, where one often observes experimentally expression levels of biomolecules such as proteins and tries to reconstruct their interaction and signalling networks [15]. Third problem is the analysis of physical systems where a graph can describe interactions between multiple objects [16]. In particular, high-energy physicists dealing with complex particle interactions have recently been showing keen interest in graph-based approaches [17]. Last but not least are problems in NLP, where graph neural networks can be seen as generalisations of the transformer architecture. Many of the mentioned problems also raise questions on incorporating priors on the graph structure, which is still largely open: for example, one may wish to force the graph to obey certain construction rules or be compatible with some statistical model [18].

Ť这里是潜在的图表学习有很多其他有趣的应用。 一种是少拍学习，其中基于图的技术可以帮助总结一些示例。 在数据标记成本显着的计算机视觉中，快速学习变得越来越重要[5]。 另一个领域是生物学，其中人们经常通过实验观察诸如蛋白质之类的生物分子的表达水平，并试图重建其相互作用和信号网络[15]。 第三个问题是对物理系统的分析，其中图形可以描述多个对象之间的相互作用[16]。 特别是，处理复杂粒子相互作用的高能物理学家最近对基于图的方法表现出了浓厚的兴趣[17]。 最后但并非最不重要的是NLP中的问题，其中图神经网络可以看作是变压器体系结构的概括 。 许多提到的问题也引发了关于将先验值合并到图结构中的问题，而图结构仍然在很大程度上是开放的：例如，人们可能希望强迫图服从某些构造规则或与某些统计模型兼容[18]。

I believe that latent graph learning, while not entirely new, offers a new perspective on old problems. It is for sure an interesting setting of graph ML problems, providing a new playground for GNN researchers.

我相信潜图学习虽然不是全新的，但它为旧问题提供了新的视角。 毫无疑问，这是一个有趣的图ML问题设置，为GNN研究人员提供了一个新的场所。

[1] A slightly different but related class of methods seeks to decouple the graph provided as input from the computational graph used for message passing in graph neural networks, see e.g. J. Halcrow et al. Grale: Designing networks for graph learning (2020). arXiv:2007.12002. There are multiple reasons why one may wish to do it, one of which is breaking the bottlenecks related to the exponential growth of the neighbourhood size in some graphs, as shown by U. Alon and E. Yahav, On the bottleneck of graph neural networks and its practical implications (2020). arXiv:2006.05205.

[1]略有不同但相关的一类方法试图将作为输入提供的图与用于图神经网络中消息传递的计算图解耦，例如参见J. Halcrow等。 Grale：设计图学习网络 (2020年)。 arXiv：2007.12002。 可能有多种原因，其中一个正在打破某些图中与邻域大小指数增长有关的瓶颈，如U. Alon和E. Yahav所著， 论图神经网络的瓶颈及其实际意义 (2020年)。 arXiv：2006.05205​​。

[2] Problems of reconstructing graphs underlying some data were considered in the signal processing context in the PhD thesis of Xiaowen Dong, in whose defence committee I took part in May 2014, just a few days before the birth of my son. X. Dong et al. Learning graphs from data: A signal representation perspective (2019), IEEE Signal Processing Magazine 36(3):44–63 presents a good summary of this line of work. A more recent incarnation of these approaches from the perspective of network games is the work of Y. Leng et al. Learning quadratic games on networks (2020). Proc. ICML, on whose PhD committee at MIT I was earlier this year.

[2] 2014年5月，就在我儿子出生几天之前，我在董晓文的国防委员会的博士学位论文中就信号处理方面考虑了重构一些数据的图形问题。 X.Dong等。 从数据中学习图表：从信号表示的角度(2019年)，IEEE信号处理杂志36(3)：44–63很好地总结了这方面的工作。 从网络游戏的角度来看，这些方法的更新版本是Y. Leng等人的工作。 在网络上学习二次游戏 (2020)。 进程 ICML，今年早些时候曾在麻省理工学院的博士学位委员会任职。

[3] C. Qi et al. PointNet: Deep learning on point sets for 3D classification and segmentation (2017), Proc. CVPR. PointNet is an architecture for deep learning on sets, where a shared function is applied to the representation of each point, and can be considered as a trivial case of a GNN applied to a graph with empty edge set.

[3] C. Qi等。 PointNet：关于3D分类和分割的点集的深度学习 (2017)，Proc。 CVPR。 PointNet是一种用于集合上深度学习的体系结构，其中将共享功能应用于每个点的表示，并且可以视为将GNN应用于带有空边缘集的图形的琐碎情况。

[4] Y. Wang et al. Dynamic graph CNN for learning on point clouds (2019). ACM Trans. Graphics 38(5):146. This paper has become quite popular in the computer graphics community and is often used as a baseline for point cloud methods. Ironically, it was rejected from SIGGRAPH in 2018 and was presented at the same conference only two years later after having gathered over 600 citations.

[4] Y. Wang等。 用于点云学习的动态图CNN (2019)。 ACM Trans。 图形38(5)：146。 本文已在计算机图形社区中变得非常流行，并且经常用作点云方法的基准。 具有讽刺意味的是，它在2018年被SIGGRAPH拒绝了，并且在收集了600多次引用之后仅在两年后的同一个会议上被提出。

[5] A. Kazi et al., Differentiable Graph Module (DGM) for graph convolutional networks (2020) arXiv:2002.04999. We show multiple applications, including medical imaging, 3D point cloud analysis, and few shot learning. See also our paper L. Cosmo et al. Latent patient network learning for automatic diagnosis (2020). Proc. MICCAI, focusing on a medical application of this method. Anees was a visiting PhD student in my group at Imperial College in 2019.

[5] A. Kazi等人， 图卷积网络的可微图模块(DGM) (2020)arXiv：2002.04999。 我们展示了多种应用，包括医学成像，3D点云分析和少量镜头学习。 另请参见我们的论文L. Cosmo等。 用于自动诊断的潜在患者网络学习 (2020)。 进程 MICCAI，专注于此方法的医学应用。 Anees于2019年在我所在的帝国理工学院(Imperial College)担任访问博士生。

[6] To the best of my knowledge, the first use of GNNs for brain disease prediction is by S. Parisot et al. Disease prediction using graph convolutional networks: application to autism spectrum disorder and Alzheimer’s disease (2017). Proc. MICCAI. The key drawback of this approach was a handcrafted construction of the graph from demographic features.

[6]据我所知，S。Parisot等人首次将GNN用于脑疾病预测。 使用图卷积网络进行疾病预测：在自闭症谱系障碍和阿尔茨海默氏病中的应用 (2017)。 进程 MICCAI。 这种方法的主要缺点是根据人口统计特征对图表进行手工构建。

[7] Formally speaking, it is not a “manifold” in the differential geometric sense of the term, since for example the local dimension can vary at different points. However, it is a convenient metaphor.

[7]从形式上讲，它不是该术语的微分几何意义上的“歧管”，因为例如局部尺寸可以在不同点变化。 但是，这是一个方便的隐喻。

[8] The more correct term is “non-Euclidean” rather than “non-linear”.

[8]更正确的术语是“非欧几里得”而不是“非线性”。

[9] J. B. Tenenbaum et al., A global geometric framework for nonlinear dimensionality reduction (2000), Science 290:2319–2323. Introduced the Isomap algorithm that embeds the data manifold by trying to preserve the geodesic distances on it, approximated using a k-NN graph. Geodesic distances on the graph are the lengths of the shortest paths connecting any pair of points, computed by means of the Dijkstra algorithm. Endowed with such a distance metric, the dataset is considered as a (non-Euclidean) metric space. A configuration of points in a low-dimensional space whose pairwise Euclidean distances are equal to the graph geodesic distances is known as isometric embedding in metric geometry. Usually, isometric embeddings do not exist and one has to resort to an approximation that preserves the distances the most in some sense. One way of computing such an approximation is by means of multidimensional scaling (MDS) algorithms.

[9] JB Tenenbaum等，非线性降维的全局几何框架 (2000年)，科学290：2319–2323。 引入了Isomap算法，该算法通过尝试保留测地距离来嵌入数据流形，使用k -NN图进行近似。 图上的测地距离是通过Dijkstra算法计算的连接任意一对点的最短路径的长度。 有了这样的距离度量，数据集被视为(非欧几里得)度量空间。 两维欧氏距离等于图形测地线距离的低维空间中的点的配置称为度量几何中的等距嵌入 。 通常情况下，不存在等距嵌入，因此必须诉诸某种近似来在某种意义上最大程度地保留距离。 一种计算这种近似值的方法是借助多维缩放 (MDS)算法。

[10] S. T. Roweis and L. K. Saul, Nonlinear dimensionality reduction by locally linear embedding (2000). Science 290:2323–2326.

[10] ST Roweis和LK Saul， 通过局部线性嵌入降低非线性维数 (2000)。 科学290：2323-2326。

[11] M, Belkin and P. Niyogi, Laplacian eigenmaps and spectral techniques for embedding and clustering (2001). Proc. NIPS.

[11] M，Belkin和P. Niyogi， 拉普拉斯特征图和嵌入和聚类的光谱技术 (2001年)。 进程 NIPS。

[12] J. Svoboda et al. PeerNets: Exploiting peer wisdom against adversarial attacks (2019), Proc. ICLR uses GNN module that aggregates information from multiple images to reduce the sensitivity of CNNs to adversarial perturbations of the input.

[12] J. Svoboda等。 PeerNets：利用同行智慧对抗对抗攻击 (2019)，Proc。 ICLR使用GNN模块来聚合来自多个图像的信息，以降低CNN对输入的对抗性扰动的敏感性。

[13] Non-local means is a non-linear image filtering technique introduced by A. Buades et al., A non-local algorithm for image denoising (2005), Proc. CVPR. It can be seen as a precursor to modern attention mechanisms used in deep learning. Non-local means itself is a variant of edge-preserving diffusion methods such as the Beltrami flow proposed by my PhD advisor Ron Kimmel in the paper R. Kimmel et al., From high energy physics to low level vision (1997), Proc. Scale-Space Theories in Computer Vision, or the bilateral filter from C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images (1998). Proc. ICCV.

[13]非局部均值是由A. Buades等人提出的一种非线性图像滤波技术， 一种图像去噪的非局部算法 (2005年)，Proc.Natl.Acad.Sci.USA，87：3877-5。 CVPR。 可以将其视为深度学习中使用的现代注意力机制的先驱。 非局部均值本身是保留边缘的扩散方法的一种变体，例如我的博士生顾问Ron Kimmel在论文R.Kimmel等人( 从高能物理学到低视力 (1997年)，Proc.Natl.Acad.Sci.USA)中提出的Beltrami流。 计算机视觉的比例空间理论，或C. Tomasi和R. Manduchi的双边过滤器，用于灰度和彩色图像的双边过滤器 (1998年)。 进程 ICCV。

[14] Adversarial perturbation is a carefully constructed input noise that significantly reduces the performance of CNNs e.g. in image classification. This phenomenon was described in C. Szegedy et al. Intriguing properties of neural networks (2014), Proc. ICLR, and resulted in multiple follow-up works that showed bizarre adversarial attacks as extreme as changing a single pixel (J. Su et al. One pixel attack for fooling deep neural networks (2017), arXiv:1710.08864) or data-independent “universal” perturbations (S. M. Moosavi-Dezfooli et al., Universal adversarial perturbations (2017), Proc. CVPR).

[14]对抗性扰动是一种精心构造的输入噪声，它会大大降低CNN的性能，例如在图像分类中。 这种现象在C. Szegedy等人中得到了描述。 神经网络的有趣属性 (2014)，过程。 ICLR，并进行了多次后续研究，这些研究表明，怪异的对抗攻击就像改变单个像素一样极端(J. Su等人。 愚弄深度神经网络的一次像素攻击 (2017年)，arXiv：1710.08864)或与数据无关的“通用”摄动(SM Moosavi-Dezfooli等人， 通用对抗摄动 (2017年)，Proc。CVPR)。

[15] Y. Yu et al., DAG-GNN: DAG structure learning with graph neural networks (2019). Proc. ICML.

[15] Y. Yu等人， DAG-GNN：使用图神经网络进行DAG结构学习 (2019年)。 进程 ICML。

[16] T. Kipf et al., Neural relational inference for interaction systems (2019). Proc. ICML. Recovers a graph “explaining” the physics of a system by using a variational autoencoder, in which the latent vectors represent the underlying interaction graph and the decoder is a graph neural network.

[16] T. Kipf等人， 交互系统的神经关系推断 (2019年)。 进程 ICML。 通过使用变分自动编码器来恢复“解释”系统物理关系的图，其中潜在矢量表示基础交互图，而解码器是图神经网络。

[17] The use of GNNs in high-energy physics is a fascinating topic worth a separate post. Together with my PhD student Federico Monti we have worked with the IceCube collaboration developing probably the first GNN-based approach for particle physics. Our paper N. Choma, F. Monti et al., Graph neural networks for IceCube signal classification (2018), Proc. ICMLA, where we used the MoNet architecture for astrophysical neutrino classification, got the best paper award. In a more recent work, S. R. Qasim et al., Learning representations of irregular particle-detector geometry with distance-weighted graph networks (2019), European Physical Journal C 79, used a variant of DGCNN similar to DGM called GravNet for particle reconstruction.

[17]在高能物理中使用GNN是一个有趣的话题，值得单独撰写。 我们与我的博士生Federico Monti一起与IceCube合作，共同开发了第一个基于GNN的粒子物理学方法。 我们的论文N.Choma，F.Monti等人，《 用于IceCube信号分类的图神经网络》 (2018年)，Proc。 我们将MoNet架构用于天体中微子分类的ICMLA获得了最佳论文奖。 在最近的工作中，SR Qasim等人的《 使用距离加权图网络学习不规则粒子检测器几何的表示》 (2019年)，《欧洲物理杂志》 C 79，使用了类似于DGM的DGCNN变种GravNet进行了粒子重建。

[18] A somewhat related class of approaches are generative graph models, see e.g. Y. Li et al, Learning deep generative models of graphs (2018). arXiv:1803.03324. One of the applications is generating molecular graphs of chemical compounds that adhere to strict construction rules.

[18]某种程度相关的方法是生成图模型，请参见例如Y. Li等人，《 学习图的深层生成模型》 (2018年)。 arXiv：1803.03324。 应用之一是生成遵守严格构造规则的化学化合物的分子图。

[19] There are many more works on latent graph learning papers that have appeared in the past couple of years — if I omit some, this is because my goal is not to be exhaustive but rather to show a principle. I will refer to one additional work of L. Franceschi et al. Learning discrete structures for graph neural networks (2019). Proc. ICML, which also mentions the relation to Isomap and manifold learning techniques.

[19]在过去几年中出现了很多有关潜图学习论文的著作-如果我省略一些，那是因为我的目标不是详尽无遗，而是要展示一个原理。 我将参考L. Franceschi等人的另一篇著作。 学习图神经网络的离散结构 (2019)。 进程 ICML，还提到了与Isomap和多种学习技术的关系。

I am grateful to Ben Chamberlain, Xiaowen Dong, Fabrizio Frasca, Anees Kazi, and Yue Wang for proof-reading this post, and to Gal Mishne for pointing to the origins of the Swiss roll. See my other posts on Medium, or follow me on Twitter.

我感谢本·张伯伦，董小文，Fabrizio Frasca，Anees Kazi和王悦对本文进行校对，并感谢Gal Mishne指出了瑞士卷的起源。 在Medium上 查看我的 其他帖子 ，或在 Twitter上 关注我 。