学习视觉数据的变换不变表示是计算机视觉中的一个重要问题。深度卷积网络已经为图像和视频分类任务提供了显着的结果。但是,它们在经历几何变换的图像分类方面仅取得了有限的成功。在这项工作中,我们提出了一种新颖的基于图形的变换图形网络(TIGraNet),它可以学习基于图形的特征​​,这些特征本质上不等同于等距变换,例如输入图像的旋转和平移。特别地,图像被表示为图上的信号,这允许用图谱分解和动态图池池替换深网络中的经典卷积和汇集层,这些层一起有助于不变等值变换。我们的实验表明,与对数据转换非常敏感的经典架构相比,来自测试集的旋转和转换图像具有高性能。我们框架的固有不变性提供了关键优势,例如通过有限的培训集提高数据可变性和持续性能。我们的代码可在线获取。
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Two architectures that generalize convolutional neu-ral networks (CNNs) for the processing of signals supported on graphs are introduced. We start with the selection graph neural network (GNN), which replaces linear time invariant filters with linear shift invariant graph filters to generate convo-lutional features and reinterprets pooling as a possibly nonlinear subsampling stage where nearby nodes pool their information in a set of preselected sample nodes. A key component of the architecture is to remember the position of sampled nodes to permit computation of convolutional features at deeper layers. The second architecture, dubbed aggregation GNN, diffuses the signal through the graph and stores the sequence of diffused components observed by a designated node. This procedure effectively aggregates all components into a stream of information having temporal structure to which the convolution and pooling stages of regular CNNs can be applied. A multinode version of aggregation GNNs is further introduced for operation in large scale graphs. An important property of selection and aggregation GNNs is that they reduce to conventional CNNs when particularized to time signals reinterpreted as graph signals in a circulant graph. Comparative numerical analyses are performed in a source localization application over synthetic and real-world networks. Performance is also evaluated for an authorship attribution problem and text category classification. Multinode aggregation GNNs are consistently the best performing GNN architecture.
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In the Multiterminal Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the min-cut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2 − 2/ k of the optimal cut weight.
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图形卷积神经网络(GCNN)是深度学习领域最新的兴奋点,它们的应用正迅速传播到多个跨领域,包括生物信息学,化学信息学,社会网络,自然语言处理和计算机视觉。在本文中,我们使用在cite {hinton2011transforming}中提供的capsuleidea来展示和解决GCNN模型的一些基本弱点,并提出我们的Graph CapsuleNetwork(GCAPS-CNN)模型。此外,我们设计了GCAPS-CNN模型来解决当前GCNN模型具有挑战性的特殊图形分类问题。通过大量实验,我们证明了我们提出的GraphCapsule网络可以明显优于现有的最先进的学习方法和图形分类基准数据集上的图形内核。
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在这项工作中,我们感兴趣的是将卷积神经网络(CNN)从低维规则网格(其中包含图像,视频和语音)推广到高维不规则域,例如社交网络,脑连接词或单词'嵌入,表示通过图表。我们在谱图理论的背景下提出了CNN的形式,它提供了必要的数学背景和有效的数值方案,以在图上设计快速局部卷积滤波器。重要的是,所提出的技术提供了与经典CNN相同的线性计算复杂度和恒定学习复杂度,同时对任何图形结构都是通用的。对MNIST和20NEWS的实验证明了这种新颖的学习系统能够学习图形上的局部,静止和组合特征。
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图形神经网络将深度神经网络模型推广到图形结构数据,近年来引起了越来越多的关注。它们通过转换,传播和聚合节点特征来学习节点表示,并且已被证明可以提高许多与图形相关的任务(如节点分类和链接预测)的性能。为了将图形网络应用于图分类任务,需要从节点表示生成\ textit {图表示}的方法。一个共同点是全局组合节点表示。然而,忽略了丰富的结构信息。因此,期望分层池化过程在图形表示学习期间保留图形结构。最近有一些关于分层学习图形表示的工作类似于传统卷积神经网络(CNN)中的池化步骤。然而,在池化过程中,局部结构信息仍然被大大忽略。在本文中,我们引入了基于图形傅里叶变换的池化运算符$ \ pooling $,它可以在池化过程中利用节点特征和局部结构。然后,我们基于池操作符设计池化层,这些层与传统的GCN卷积层进一步组合,形成图形分类的图形神经网络框架$ \ m $。提供理论分析以了解来自本地和全球视角的$ \ pooling $。 $ 6 $常用基准测试的图分类任务的实验结果证明了所提出的框架的有效性。
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最近,许多研究人员一直专注于图形的神经网络的定义。许多这些方法的基本组成部分仍然是近十年前提出的图形卷积思想。在本文中,我们扩展了这个基本组件,遵循从众所周知的多维张量上的卷积滤波器得到的直觉。特别地,我们推导出一种简单,有效和有效的方法来引入影响滤波器大小的图形卷积的超参数,即其在所考虑的图上的感知场。我们用真实世界图形数据集的实验结果表明,所提出的图卷积滤波器改进了深度图卷积网络的预测性能。
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最近的图神经网络基于在谱域中操作的多项式滤波器实现卷积层。在本文中,我们提出了一种基于自回归移动平均(ARMA)滤波器的新型图卷积层,与多项式滤波器相比,由于丰富的传递函数可以解释状态的概念,因此可以提供更灵活的响应。我们使用递归和分布式构造实现ARMA过滤器,获得有效训练的卷积层,它在节点空间中定位并且可以应用于具有不同拓扑的图形。为了学习更多抽象和压缩表示网络的更深层,我们基于nodedecimation替换原始图的粗化版本上的卷积的池操作。我们考虑三个主要的图形推断问题:半监督节点分类,图形分类和图形信号分类。结果表明,所提出的具有ARMA滤波器的网络优于基于多项式滤波器的网络,并在若干任务中定义了新的最新技术。
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In this paper, we develop a measure-theoretic version of the junction tree algorithm to compute desired marginals of a product function. We reformulate the problem in a measure-the-oretic framework, where the desired marginals are viewed as corresponding conditional expectations of a product of random variables. We generalize the notions of independence and junction trees to collections of-fields on a space with a signed measure. We provide an algorithm to find such a junction tree when one exists. We also give a general procedure to augment the-fields to create in-dependencies, which we call "lifting." This procedure is the counterpart of the moralization and triangulation procedure in the conventional generalized distributive law (GDL) framework, in order to guarantee the existence of a junction tree. Our procedure includes the conventional GDL procedure as a special case. However, it can take advantage of structures at the atomic level of the sample space to produce junction tree-based algorithms for computing the desired marginals that are less complex than those GDL can discover , as we argue through examples. Our formalism gives a new way by which one can hope to find low-complexity algorithms for marginalization problems.
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In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.
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Centrality indices
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We generalize the scattering transform to graphs and consequently construct a convolutional neural network on graphs. We show that under certain conditions, any feature generated by such a network is approximately invariant to permutations and stable to graph manipulations. Numerical results demonstrate competitive performance on relevant datasets.
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In this work we propose the construction of two-channel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filterbanks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to "multi-dimensional" separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations.
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In social settings, individuals interact through webs of relationships. Eachindividual is a node in a complex network (or graph) of interdependencies andgenerates data, lots of data. We label the data by its source, or formallystated, we index the data by the nodes of the graph. The resulting signals(data indexed by the nodes) are far removed from time or image signals indexedby well ordered time samples or pixels. DSP, discrete signal processing,provides a comprehensive, elegant, and efficient methodology to describe,represent, transform, analyze, process, or synthesize these well ordered timeor image signals. This paper extends to signals on graphs DSP and its basictenets, including filters, convolution, z-transform, impulse response, spectralrepresentation, Fourier transform, frequency response, and illustrates DSP ongraphs by classifying blogs, linear predicting and compressing data fromirregularly located weather stations, or predicting behavior of customers of amobile service provider.
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Kleinberg引入了三种自然聚类属性或公理,并且显示它们不能通过任何聚类算法同时满足。我们提出了一种新的聚类属性Monotonic Consistency,它避免了Kleinberg一致性公理中众所周知的问题行为,以及不可能性结果。也就是说,我们描述了一种聚类算法,MorseClustering,受Morse Theory在差分拓扑中的启发,它满足了Kininberg原始公理,其一致性被单调一致性所取代.Morse聚类揭示了集合或图形上的基础流动结构,并返回表示代表吸引盆的树的分区关键的转移。我们还概括了Kleinberg对稀疏图的公理化方法,显示了一致性的不可能结果,以及单调一致性和莫尔斯聚类的可能性结果。
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Finding the maximum a posteriori (MAP) assignment of a discrete-state distribution specified by a graphical model requires solving an integer program. The max-product algorithm, also known as the max-plus or min-sum algorithm, is an iterative method for (approximately) solving such a problem on graphs with cycles. We provide a novel perspective on the algorithm, which is based on the idea of reparameterizing the distribution in terms of so-called pseudo-max-marginals on nodes and edges of the graph. This viewpoint provides conceptual insight into the max-product algorithm in application to graphs with cycles. First, we prove the existence of max-product fixed points for positive distributions on arbitrary graphs. Next, we show that the approximate max-marginals computed by max-product are guaranteed to be consistent, in a suitable sense to be defined, over every tree of the graph. We then turn to characterizing the nature of the approximation to the MAP assignment computed by max-product. We generalize previous work by showing that for any graph, the max-product assignment satisfies a particular optimality condition with respect to any subgraph containing at most one cycle per connected component. We use this optimality condition to derive upper bounds on the difference between the log probability of the true MAP assignment, and the log probability of a max-product assignment. Finally, we consider extensions of the max-product algorithm that operate over higher-order cliques, and show how our reparameterization analysis extends in a natural manner.
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We study the problem of finding shortest tours/paths for "lawn mowing" and "milling" problems: Given a region in the plane, and given the shape of a "cutter" (typically, a circle or a square), find a shortest tour/path for the cutter such that every point within the region is covered by the cutter at some position along the tour/path. In the milling version of the problem, the cutter is constrained to stay within the region. The milling problem arises naturally in the area of automatic tool path generation for NC pocket machining. The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general. We give efficient constant-factor approximation algorithms for both problems. In particular, we give a (3 + ε)-approximation algorithm for the lawn mowing problem and a 2.5-approximation algorithm for the milling problem. Furthermore, we give a simple 6 5-approximation algorithm for the TSP problem in simple grid graphs, which leads to an 11 5-approximation algorithm for milling simple rectilinear polygons.
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Motivated by applications in network epidemiology, we consider the problem of determining whether it is possible to delete at most k edges from a given input graph (of small treewidth) so that the resulting graph avoids a set F of forbidden subgraphs; of particular interest is the problem of determining whether it is possible to delete at most k edges so that the resulting graph has no connected component of more than h vertices, as this bounds the worst-case size of an epidemic. While even this special case of the problem is NP-complete in general (even when h = 3), we provide evidence that many of the real-world networks of interest are likely to have small treewidth, and we describe an algorithm which solves the general problem in time 2 O(|F |w r) n on an input graph having n vertices and whose treewidth is bounded by a fixed constant w, if each of the subgraphs we wish to avoid has at most r vertices. For the special case in which we wish only to ensure that no component has more than h vertices, we improve on this to give an algorithm running in time O((wh) 2w n), which we have implemented and tested on real datasets based on cattle movements.
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In this paper, we consider the inverse spanning tree problem. Given an undi-0 Ž 0 0. rected graph G s N , A with n nodes, m arcs, an arc cost vector c, and a spanning tree T 0 , the inverse spanning tree problem is to perturb the arc cost vector c to a vector d so that T 0 is a minimum spanning tree with respect to the < < < < cost vector d and the cost of perturbation given by d y c s Ý d y c is Ž i, j.g A ij ij minimum. We show that the dual of the inverse spanning tree problem is a Ž bipartite node weighted matching problem on a specially structured graph which. Ž .Ž. we call the path graph that contains m nodes and as many as m y n q 1 n y 1 Ž. s O nm arcs. We first transform the bipartite node weighted matching problem into a specially structured minimum cost flow problem and use its special structure Ž 3. to develop an O n algorithm. We next use its special structure more effectively Ž 2. Ž 3. and develop an O n log n time algorithm. This improves the previous O n time Ž. algorithm due to Sokkalingam et al.
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