Transferability of Spectral Graph Convolutional Neural Networks
Ludwig Maximilian University of Munich（慕尼黑路德维希-马克西米利安大学）；Università della Svizzera italiana（瑞士意大利语区大学）；University of Tromsø（特罗姆瑟大学）；Imperial College London（伦敦帝国理工学院）；Twitter（推特）
Ron Levie, Wei Huang, Lorenzo Bucci, Michael M. Bronstein, Gitta Kutyniok
This paper focuses on spectral graph convolutional neural networks (ConvNets), where filters are defined as elementwise multiplication in the frequency domain of a graph. In machine learning settings where the dataset consists of signals defined on many different graphs, the trained ConvNet should generalize to signals on graphs unseen in the training set. It is thus important to transfer ConvNets between graphs. Transferability, which is a certain type of generalization capability, can be loosely defined as follows: if two graphs describe the same phenomenon, then a single filter or ConvNet should have similar repercussions on both graphs. This paper aims at debunking the common misconception that spectral filters are not transferable. We show that if two graphs discretize the same "continuous" space, then a spectral filter or ConvNet has approximately the same repercussion on both graphs. Our analysis is more permissive than the standard analysis. Transferability is typically described as the robustness of the filter to small graph perturbations and re-indexing of the vertices. Our analysis accounts also for large graph perturbations. We prove transferability between graphs that can have completely different dimensions and topologies, only requiring that both graphs discretize the same underlying space in some generic sense.
Graph Neural Networks (GNNs) are information processing architectures for signals supported on graphs. They are presented here as generalizations of convolutional neural networks (CNNs) in which individual layers contain banks of graph convolutional filters instead of banks of classical convolutional filters. Otherwise, GNNs operate as CNNs. Filters are composed with pointwise nonlinearities and stacked in layers. It is shown that GNN architectures exhibit equivariance to permutation and stability to graph deformations. These properties help explain the good performance of GNNs that can be obs…
This paper focuses on spectral filters on graphs, namely filters defined as elementwise multiplication in the frequency domain of a graph. In many graph signal processing settings, it is important to transfer a filter from one graph to another. One example is in graph convolutional neural networks (ConvNets), where the dataset consists of signals defined on many different graphs, and the learned filters should generalize to signals on new graphs, not present in the training set. A necessary condition for transferability (the ability to transfer filters) is stability. Namely, given a graph filt…
The Transferability of Downsamped Sparse Graph Convolutional Networks
School of Information Science and Technology, Fudan University（复旦大学信息科学与技术学院）；State Key Laboratory of Integrated Chips and Systems, Fudan University（复旦大学集成芯片与系统国家重点实验室）；School of Electrical and Electronic Engineering, Nanyang Technological University（南洋理工大学电气与电子工程学院）
University of Pennsylvania（宾夕法尼亚大学）；Dept. of Electrical and Systems Eng.（电气与系统工程系）；Philadelphia, PA 19143（费城，PA 19143）
Graph neural networks (GNNs) rely on graph convolutions to extract local features from network data. These graph convolutions combine information from adjacent nodes using coefficients that are shared across all nodes. Since these coefficients are shared and do not depend on the graph, one can envision using the same coefficients to define a GNN on another graph. This motivates analyzing the transferability of GNNs across graphs. In this paper we introduce graphon NNs as limit objects of GNNs and prove a bound on the difference between the output of a GNN and its limit graphon-NN. This bound v…
Spectral graph convolutional networks are generalizations of standard convolutional networks for graph-structured data using the Laplacian operator. A common misconception is the instability of spectral filters, i.e. the impossibility to transfer spectral filters between graphs of variable size and topology. This misbelief has limited the development of spectral networks for multi-graph tasks in favor of spatial graph networks. However, recent works have proved the stability of spectral filters under graph perturbation. Our work complements and emphasizes further the high quality of spectral t…
Modern deep convolutional networks (CNNs) are often criticized for not generalizing under distributional shifts. However, several recent breakthroughs in transfer learning suggest that these networks can cope with severe distribution shifts and successfully adapt to new tasks from a few training examples. In this work we study the interplay between out-of-distribution and transfer performance of modern image classification CNNs for the first time and investigate the impact of the pre-training data size, the model scale, and the data preprocessing pipeline. We find that increasing both the trai…
X-ray imagery security screening is essential to maintaining transport security against a varying profile of threat or prohibited items. Particular interest lies in the automatic detection and classification of weapons such as firearms and knives within complex and cluttered X-ray security imagery. Here, we address this problem by exploring various end-to-end object detection Convolutional Neural Network (CNN) architectures. We evaluate several leading variants spanning the Faster R-CNN, Mask R-CNN, and RetinaNet architectures to explore the transferability of such models between varying X-ray…
Training deep neural networks often requires careful hyper-parameter tuning and significant computational resources. In this paper, we propose ConvTimeNet (CTN): an off-the-shelf deep convolutional neural network (CNN) trained on diverse univariate time series classification (TSC) source tasks. Once trained, CTN can be easily adapted to new TSC target tasks via a small amount of fine-tuning using labeled instances from the target tasks. We note that the length of convolutional filters is a key aspect when building a pre-trained model that can generalize to time series of different lengths acro…
LMU Munich（慕尼黑大学）；Technion - Israel Institute of Technology（以色列理工学院）；University of Tromsø（特罗姆瑟大学）
We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability: the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as th…
Automatic detection of prohibited items within complex and cluttered X-ray security imagery is essential to maintaining transport security, where prior work on automatic prohibited item detection focus primarily on pseudo-colour (rgb}) X-ray imagery. In this work we study the impact of variant X-ray imagery, i.e., X-ray energy response (high, low}) and effective-z compared to rgb, via the use of deep Convolutional Neural Networks (CNN) for the joint object detection and segmentation task posed within X-ray baggage security screening. We evaluate state-of-the-art CNN architectures (Mask R-CNN, …
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；School of Cyber Security, University of Chinese Academy of Sciences（中国科学院大学网络空间安全学院）
Recently, non-stationary spectral kernels have drawn much attention, owing to its powerful feature representation ability in revealing long-range correlations and input-dependent characteristics. However, non-stationary spectral kernels are still shallow models, thus they are deficient to learn both hierarchical features and local interdependence. In this paper, to obtain hierarchical and local knowledge, we build an interpretable convolutional spectral kernel network (\texttt{CSKN}) based on the inverse Fourier transform, where we introduce deep architectures and convolutional filters into no…
In this work, we are interested in generalizing convolutional neural networks (CNNs) from low-dimensional regular grids, where image, video and speech are represented, to high-dimensional irregular domains, such as social networks, brain connectomes or words' embedding, represented by graphs. We present a formulation of CNNs in the context of spectral graph theory, which provides the necessary mathematical background and efficient numerical schemes to design fast localized convolutional filters on graphs. Importantly, the proposed technique offers the same linear computational complexity and c…
Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）
Discrete Fourier transforms provide a significant speedup in the computation of convolutions in deep learning. In this work, we demonstrate that, beyond its advantages for efficient computation, the spectral domain also provides a powerful representation in which to model and train convolutional neural networks (CNNs). We employ spectral representations to introduce a number of innovations to CNN design. First, we propose spectral pooling, which performs dimensionality reduction by truncating the representation in the frequency domain. This approach preserves considerably more information per …
Sabanci University（萨班哲大学）；University of Padua（帕多瓦大学）；Moulay Ismail University of Meknes（梅克内斯穆莱·伊斯梅尔大学）；Florida International University（佛罗里达国际大学）
Convolutional Neural Networks (CNNs) models are one of the most frequently used deep learning networks, and extensively used in both academia and industry. Recent studies demonstrated that adversarial attacks against such models can maintain their effectiveness even when used on models other than the one targeted by the attacker. This major property is known as transferability, and makes CNNs ill-suited for security applications. In this paper, we provide the first comprehensive study which assesses the robustness of CNN-based models for computer networks against adversarial transferability. F…
Helsinki Institute for Information Technology, HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）
We introduce the convolutional spectral kernel (CSK), a novel family of non-stationary, nonparametric covariance kernels for Gaussian process (GP) models, derived from the convolution between two imaginary radial basis functions. We present a principled framework to interpret CSK, as well as other deep probabilistic models, using approximated Fourier transform, yielding a concise representation of input-frequency spectrogram. Observing through the lens of the spectrogram, we provide insight on the interpretability of deep models. We then infer the functional hyperparameters using scalable vari…
Virginia Tech（弗吉尼亚理工大学）；Tsinghua University（清华大学）；University of California at Berkeley（加州大学伯克利分校）
Graph convolutional networks (GCNs) are powerful tools for graph-structured data. However, they have been recently shown to be vulnerable to topological attacks. To enhance adversarial robustness, we go beyond spectral graph theory to robust graph theory. By challenging the classical graph Laplacian, we propose a new convolution operator that is provably robust in the spectral domain and is incorporated in the GCN architecture to improve expressivity and interpretability. By extending the original graph to a sequence of graphs, we also propose a robust training paradigm that encourages transfe…
In convolutional neural networks, the linear transformation of multi-channel two-dimensional convolutional layers with linear convolution is a block matrix with doubly Toeplitz blocks. Although a "wrapping around" operation can transform linear convolution to a circular one, by which the singular values can be approximated with reduced computational complexity by those of a block matrix with doubly circulant blocks, the accuracy of such an approximation is not guaranteed. In this paper, we propose to inspect such a linear transformation matrix through its asymptotic spectral representation - t…
University of Pennsylvania（宾夕法尼亚大学）；Duke University（杜克大学）
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 …
Indian Institute of Space Science and Technology（印度空间科学与技术研究所）
Graph convolutional neural networks (GCNNs) have been widely used in graph learning. It has been observed that the smoothness functional on graphs can be defined in terms of the graph Laplacian. This fact points out in the direction of using Laplacian in deriving regularization operators on graphs and its consequent use with spectral GCNN filter designs. In this work, we explore the regularization properties of graph Laplacian and proposed a generalized framework for regularized filter designs in spectral GCNNs. We found that the filters used in many state-of-the-art GCNNs can be derived as a …
National University of Singapore（新加坡国立大学）；University of Wisconsin（威斯康星大学）；N.1 Institute for Health（N.1健康研究所）
This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We then define spectral filters via the LB operator on a graph. We explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters and define an update of the LB operator for pooling in the LBCNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and demonstrate the use of the proposed LB-CNN. Based on the…
Graph convolutional networks(GCNs) have become the most popular approaches for graph data in these days because of their powerful ability to extract features from graph. GCNs approaches are divided into two categories, spectral-based and spatial-based. As the earliest convolutional networks for graph data, spectral-based GCNs have achieved impressive results in many graph related analytics tasks. However, spectral-based models cannot directly work on directed graphs. In this paper, we propose an improved spectral-based GCN for the directed graph by leveraging redefined Laplacians to improve it…
In recent years, convolutional neural networks (CNNs) have been widely used by researchers to perform forensic tasks such as image tampering detection. At the same time, adversarial attacks have been developed that are capable of fooling CNN-based classifiers. Understanding the transferability of adversarial attacks, i.e. an attacks ability to attack a different CNN than the one it was trained against, has important implications for designing CNNs that are resistant to attacks. While attacks on object recognition CNNs are believed to be transferrable, recent work by Barni et al. has shown that…
Convolutional neural networks (CNNs) have achieved breakthrough performances in a wide range of applications including image classification, semantic segmentation, and object detection. Previous research on characterizing the generalization ability of neural networks mostly focuses on fully connected neural networks (FNNs), regarding CNNs as a special case of FNNs without taking into account the special structure of convolutional layers. In this work, we propose a tighter generalization bound for CNNs by exploiting the sparse and permutation structure of its weight matrices. As the generalizat…
Spectral graph convolutional neural networks (CNNs) require approximation to the convolution to alleviate the computational complexity, resulting in performance loss. This paper proposes the topology adaptive graph convolutional network (TAGCN), a novel graph convolutional network defined in the vertex domain. We provide a systematic way to design a set of fixed-size learnable filters to perform convolutions on graphs. The topologies of these filters are adaptive to the topology of the graph when they scan the graph to perform convolution. The TAGCN not only inherits the properties of convolut…
Technische Universität Berlin（柏林工业大学）；Universitá della Svizzera italiana（瑞士意大利语区大学）；Nanyang Technological University（南洋理工大学）；Imperial College London（帝国理工学院）；Intel Perceptual Computing（英特尔感知计算部门）
The rise of graph-structured data such as social networks, regulatory networks, citation graphs, and functional brain networks, in combination with resounding success of deep learning in various applications, has brought the interest in generalizing deep learning models to non-Euclidean domains. In this paper, we introduce a new spectral domain convolutional architecture for deep learning on graphs. The core ingredient of our model is a new class of parametric rational complex functions (Cayley polynomials) allowing to efficiently compute spectral filters on graphs that specialize on frequency…
Università della Svizzera Italiana（瑞士意大利语区大学）；University of Manitoba（曼尼托巴大学）；University of Exeter（埃克塞特大学）；Politecnico di Milano（米兰理工大学）
Popular graph neural networks implement convolution operations on graphs based on polynomial spectral filters. In this paper, we propose a novel graph convolutional layer inspired by the auto-regressive moving average (ARMA) filter that, compared to polynomial ones, provides a more flexible frequency response, is more robust to noise, and better captures the global graph structure. We propose a graph neural network implementation of the ARMA filter with a recursive and distributed formulation, obtaining a convolutional layer that is efficient to train, localized in the node space, and can be t…
Medical Center - University of Freiburg（弗莱堡大学医学中心）
Recently, there is increasing interest and research on the interpretability of machine learning models, for example how they transform and internally represent EEG signals in Brain-Computer Interface (BCI) applications. This can help to understand the limits of the model and how it may be improved, in addition to possibly provide insight about the data itself. Schirrmeister et al. (2017) have recently reported promising results for EEG decoding with deep convolutional neural networks (ConvNets) trained in an end-to-end manner and, with a causal visualization approach, showed that they learn to…
Deep learning methods are used on spectroscopic data to predict drug content in tablets from near infrared (NIR) spectra. Using convolutional neural networks (CNNs), features are ex- tracted from the spectroscopic data. Extended multiplicative scatter correction (EMSC) and a novel spectral data augmentation method are benchmarked as preprocessing steps. The learned models perform better or on par with hypothetical optimal partial least squares (PLS) models for all combinations of preprocessing. Data augmentation with subsequent EMSC in combination gave the best results. The deep learning model…
Child Mind Institute（儿童心智研究所）；Electrical Geodesics Inc.（电气测地公司）
The success of deep learning in computer vision has inspired the scientific community to explore new analysis methods. Within the field of neuroscience, specifically in electrophysiological neuroimaging, researchers are starting to explore leveraging deep learning to make predictions on their data without extensive feature engineering. This paper compares deep learning using minimally processed EEG raw data versus deep learning using EEG spectral features using two different deep convolutional neural architectures. One of them from Putten et al. (2018) is tailored to process raw data; the othe…
Comprehensive Analysis of Network Robustness Evaluation Based on Convolutional Neural Networks with Spatial Pyramid Pooling
School of Systems Science and Engineering, Sun Yat-sen University（中山大学系统科学与工程学院）；School of Cyber Science and Technology, University of Science and Technology of China（中国科学技术大学网络空间安全学院）
Facebook AI Research（脸书人工智能研究院）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）
Convolutional neural networks trained without supervision come close to matching performance with supervised pre-training, but sometimes at the cost of an even higher number of parameters. Extracting subnetworks from these large unsupervised convnets with preserved performance is of particular interest to make them less computationally intensive. Typical pruning methods operate during training on a task while trying to maintain the performance of the pruned network on the same task. However, in self-supervised feature learning, the training objective is agnostic on the representation transfera…
Weizmann Institute of Science（魏茨曼科学研究所）；University of Maryland at College Park（马里兰大学帕克分校）
We study the properties of various over-parametrized convolutional neural architectures through their respective Gaussian process and neural tangent kernels. We prove that, with normalized multi-channel input and ReLU activation, the eigenfunctions of these kernels with the uniform measure are formed by products of spherical harmonics, defined over the channels of the different pixels. We next use hierarchical factorizable kernels to bound their respective eigenvalues. We show that the eigenvalues decay polynomially, quantify the rate of decay, and derive measures that reflect the composition …
OPPO US Research Center（OPPO美国研究中心）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；New York University（纽约大学）；Harvard University（哈佛大学）
The memory consumption of most Convolutional Neural Network (CNN) architectures grows rapidly with increasing depth of the network, which is a major constraint for efficient network training on modern GPUs with limited memory, embedded systems, and mobile devices. Several studies show that the feature maps (as generated after the convolutional layers) are the main bottleneck in this memory problem. Often, these feature maps mimic natural photographs in the sense that their energy is concentrated in the spectral domain. Although embedding CNN architectures in the spectral domain is widely explo…
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Florida State University（佛罗里达州立大学）；Jilin University（吉林大学）
Cross-domain recommendation can alleviate the data sparsity problem in recommender systems. To transfer the knowledge from one domain to another, one can either utilize the neighborhood information or learn a direct mapping function. However, all existing methods ignore the high-order connectivity information in cross-domain recommendation area and suffer from the domain-incompatibility problem. In this paper, we propose a \textbf{J}oint \textbf{S}pectral \textbf{C}onvolutional \textbf{N}etwork (JSCN) for cross-domain recommendation. JSCN will simultaneously operate multi-layer spectral convol…
Norwegian University of Science and Technology (NTNU)（挪威科技大学（NTNU））；Tohoku University（东北大学）
Convolutional neural networks (CNNs) are widely used for image recognition and text analysis, and have been suggested for application on one-dimensional data as a way to reduce the need for pre-processing steps. Pre-processing is an integral part of multivariate analysis, but determination of the optimal pre-processing methods can be time-consuming due to the large number of available methods. In this work, the performance of a CNN was investigated for classification and regression analysis of spectral data. The CNN was compared with various other chemometric methods, including support vector …
School of Computer Science, China University of Geosciences（中国地质大学计算机学院）
In the proposed SEHybridSN model, a dense block was used to reuse shallow feature and aimed at better exploiting hierarchical spatial spectral feature. Subsequent depth separable convolutional layers were used to discriminate the spatial information. Further refinement of spatial spectral features was realized by the channel attention method, which were performed behind every 3D convolutional layer and every 2D convolutional layer. Experiment results indicate that our proposed model learn more discriminative spatial spectral features using very few training data. SEHybridSN using only 0.05 and…
Peking University（北京大学）；Bytedance Inc.（字节跳动公司）；University of Oxford（牛津大学）
We identify and overcome two key obstacles in extending the success of BERT-style pre-training, or the masked image modeling, to convolutional networks (convnets): (i) convolution operation cannot handle irregular, random-masked input images; (ii) the single-scale nature of BERT pre-training is inconsistent with convnet's hierarchical structure. For (i), we treat unmasked pixels as sparse voxels of 3D point clouds and use sparse convolution to encode. This is the first use of sparse convolution for 2D masked modeling. For (ii), we develop a hierarchical decoder to reconstruct images from multi…
Graph Convolutional Networks (GCNs) have been shown to be a powerful concept that has been successfully applied to a large variety of tasks across many domains over the past years. In this work we study the theory that paved the way to the definition of GCN, including related parts of classical graph theory. We also discuss and experimentally demonstrate key properties and limitations of GCNs such as those caused by the statistical dependency of samples, introduced by the edges of the graph, which causes the estimates of the full gradient to be biased. Another limitation we discuss is the nega…
University of Lugano（卢加诺大学）；Harvard University（哈佛大学）；Tel Aviv University（特拉维夫大学）
Deep learning on graphs and in particular, graph convolutional neural networks, have recently attracted significant attention in the machine learning community. Many of such techniques explore the analogy between the graph Laplacian eigenvectors and the classical Fourier basis, allowing to formulate the convolution as a multiplication in the spectral domain. One of the key drawback of spectral CNNs is their explicit assumption of an undirected graph, leading to a symmetric Laplacian matrix with orthogonal eigendecomposition. In this work we propose MotifNet, a graph CNN capable of dealing with…
