Variants of Graph Neural Networks (GNNs) for representation learning have been proposed recently and achieved fruitful results in various fields. Among them, Graph Attention Network (GAT) first employs a self-attention strategy to learn attention weights for each edge in the spatial domain. However, learning the attentions over edges can only focus on the local information of graphs and greatly increases the computational costs. In this paper, we first introduce the attention mechanism in the spectral domain of graphs and present Spectral Graph Attention Network (SpGAT) that learns representations for different frequency components regarding weighted filters and graph wavelets bases. In this way, SpGAT can better capture global patterns of graphs in an efficient manner with much fewer learned parameters than that of GAT. Further, to reduce the computational cost of SpGAT brought by the eigen-decomposition, we propose a fast approximation variant SpGAT-Cheby. We thoroughly evaluate the performance of SpGAT and SpGAT-Cheby in semi-supervised node classification tasks and verify the effectiveness of the learned attentions in the spectral domain.
Georgia Institute of Technology（佐治亚理工学院）；IQVIA（艾昆纬）；University of Illinois, Urbana-Champagne（伊利诺伊大学厄巴纳-香槟分校）
The attention mechanism has demonstrated superior performance for inference over nodes in graph neural networks (GNNs), however, they result in a high computational burden during both training and inference. We propose FastGAT, a method to make attention based GNNs lightweight by using spectral sparsification to generate an optimal pruning of the input graph. This results in a per-epoch time that is almost linear in the number of graph nodes as opposed to quadratic. We theoretically prove that spectral sparsification preserves the features computed by the GAT model, thereby justifying our algo…
Our goal is to efficiently compute low-dimensional latent coordinates for nodes in an input graph -- known as graph embedding -- for subsequent data processing such as clustering. Focusing on finite graphs that are interpreted as uniform samples on continuous manifolds (called manifold graphs), we leverage existing fast extreme eigenvector computation algorithms for speedy execution. We first pose a generalized eigenvalue problem for sparse matrix pair $(\A,\B)$, where $\A = Ł- μ\Q + ε\I$ is a sum of graph Laplacian $Ł$ and disconnected two-hop difference matrix $\Q$. Eigenvector $\v$ minimizi…
State-of-the-art transformer models use pairwise dot-product based self-attention, which comes at a computational cost quadratic in the input sequence length. In this paper, we investigate the global structure of attention scores computed using this dot product mechanism on a typical distribution of inputs, and study the principal components of their variation. Through eigen analysis of full attention score matrices, as well as of their individual rows, we find that most of the variation among attention scores lie in a low-dimensional eigenspace. Moreover, we find significant overlap between t…
University Politehnica of Bucharest（布加勒斯特理工大学）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利理工学院）
We investigate numerically efficient approximations of eigenspaces associated to symmetric and general matrices. The eigenspaces are factored into a fixed number of fundamental components that can be efficiently manipulated (we consider extended orthogonal Givens or scaling and shear transformations). The number of these components controls the trade-off between approximation accuracy and the computational complexity of projecting on the eigenspaces. We write minimization problems for the single fundamental components and provide closed-form solutions. Then we propose algorithms that iterative…
EPFL, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We focus in this work on the estimation of the first $k$ eigenvectors of any graph Laplacian using filtering of Gaussian random signals. We prove that we only need $k$ such signals to be able to exactly recover as many of the smallest eigenvectors, regardless of the number of nodes in the graph. In addition, we address key issues in implementing the theoretical concepts in practice using accurate approximated methods. We also propose fast algorithms both for eigenspace approximation and for the determination of the $k$th smallest eigenvalue $λ_k$. The latter proves to be extremely efficient un…
Graph Convolutional Networks (GCNs) have proven to be successful tools for semi-supervised classification on graph-based datasets. We propose a new GCN variant whose three-part filter space is targeted at dense graphs. Examples include Gaussian graphs for 3D point clouds with an increased focus on non-local information, as well as hypergraphs based on categorical data. These graphs differ from the common sparse benchmark graphs in terms of the spectral properties of their graph Laplacian. Most notably we observe large eigengaps, which are unfavorable for popular existing GCN architectures. Our…
Mila - Université de Montréal（米拉 - 蒙特利尔大学）；Facebook AI Research（脸书人工智能研究院）；CIFAR（加拿大高等研究院）
Optimization algorithms that leverage gradient covariance information, such as variants of natural gradient descent (Amari, 1998), offer the prospect of yielding more effective descent directions. For models with many parameters, the covariance matrix they are based on becomes gigantic, making them inapplicable in their original form. This has motivated research into both simple diagonal approximations and more sophisticated factored approximations such as KFAC (Heskes, 2000; Martens & Grosse, 2015; Grosse & Martens, 2016). In the present work we draw inspiration from both to propose a novel a…
Idiap Research Institute（IDIAP研究所）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of Geneva（日内瓦大学）
Transformers have been proven a successful model for a variety of tasks in sequence modeling. However, computing the attention matrix, which is their key component, has quadratic complexity with respect to the sequence length, thus making them prohibitively expensive for large sequences. To address this, we propose clustered attention, which instead of computing the attention for every query, groups queries into clusters and computes attention just for the centroids. To further improve this approximation, we use the computed clusters to identify the keys with the highest attention per query an…
Idiap Research Institute（伊迪亚普研究所）；EPFL（洛桑联邦理工学院）；University of Washington（华盛顿大学）；University of Geneva（日内瓦大学）
Transformers achieve remarkable performance in several tasks but due to their quadratic complexity, with respect to the input's length, they are prohibitively slow for very long sequences. To address this limitation, we express the self-attention as a linear dot-product of kernel feature maps and make use of the associativity property of matrix products to reduce the complexity from $\mathcal{O}\left(N^2\right)$ to $\mathcal{O}\left(N\right)$, where $N$ is the sequence length. We show that this formulation permits an iterative implementation that dramatically accelerates autoregressive transfo…
The attention module, which is a crucial component in Transformer, cannot scale efficiently to long sequences due to its quadratic complexity. Many works focus on approximating the dot-then-exponentiate softmax function in the original attention, leading to sub-quadratic or even linear-complexity Transformer architectures. However, we show that these methods cannot be applied to more powerful attention modules that go beyond the dot-then-exponentiate style, e.g., Transformers with relative positional encoding (RPE). Since in many state-of-the-art models, relative positional encoding is used as…
In this paper, we introduce an algorithm for performing spectral clustering efficiently. Spectral clustering is a powerful clustering algorithm that suffers from high computational complexity, due to eigen decomposition. In this work, we first build the adjacency matrix of the corresponding graph of the dataset. To build this matrix, we only consider a limited number of points, called landmarks, and compute the similarity of all data points with the landmarks. Then, we present a definition of the Laplacian matrix of the graph that enable us to perform eigen decomposition efficiently, using a d…
Attention-based deep networks have been successfully applied on textual data in the field of NLP. However, their application on protein sequences poses additional challenges due to the weak semantics of the protein words, unlike the plain text words. These unexplored challenges faced by the standard attention technique include (i) vanishing attention score problem and (ii) high variations in the attention distribution. In this regard, we introduce a novel λ-scaled attention technique for fast and efficient modeling of the protein sequences that addresses both the above problems. This is used t…
Large language models have become increasingly difficult to train because of the growing computation time and cost. In this work, we present SRU++, a highly-efficient architecture that combines fast recurrence and attention for sequence modeling. SRU++ exhibits strong modeling capacity and training efficiency. On standard language modeling tasks such as Enwik8, Wiki-103 and Billion Word datasets, our model obtains better bits-per-character and perplexity while using 3x-10x less training cost compared to top-performing Transformer models. For instance, our model achieves a state-of-the-art resu…
Autoregressive Transformers are strong language models but incur O(T) complexity during per-token generation due to the self-attention mechanism. Recent work proposes kernel-based methods to approximate causal self-attention by replacing it with recurrent formulations with various update rules and feature maps to achieve O(1) time and memory complexity. We explore these approaches and find that they are unnecessarily complex, and propose a simple alternative - decaying fast weights - that runs fast on GPU, outperforms prior methods, and retains 99% of attention's performance for GPT-2. We also…
The softmax content-based attention mechanism has proven to be very beneficial in many applications of recurrent neural networks. Nevertheless it suffers from two major computational limitations. First, its computations for an attention lookup scale linearly in the size of the attended sequence. Second, it does not encode the sequence into a fixed-size representation but instead requires to memorize all the hidden states. These two limitations restrict the use of the softmax attention mechanism to relatively small-scale applications with short sequences and few lookups per sequence. In this wo…
School of Integrated Technology, Yonsei University（延世大学综合技术学院）
We introduce Monte-Carlo Attention (MCA), a randomized approximation method for reducing the computational cost of self-attention mechanisms in Transformer architectures. MCA exploits the fact that the importance of each token in an input sequence varies with respect to their attention scores; thus, some degree of error can be tolerable when encoding tokens with low attention. Using approximate matrix multiplication, MCA applies different error bounds to encode input tokens such that those with low attention scores are computed with relaxed precision, whereas errors of salient elements are min…
Stanford University（斯坦福大学）；University at Buffalo, SUNY（纽约州立大学布法罗分校）
Transformers are slow and memory-hungry on long sequences, since the time and memory complexity of self-attention are quadratic in sequence length. Approximate attention methods have attempted to address this problem by trading off model quality to reduce the compute complexity, but often do not achieve wall-clock speedup. We argue that a missing principle is making attention algorithms IO-aware -- accounting for reads and writes between levels of GPU memory. We propose FlashAttention, an IO-aware exact attention algorithm that uses tiling to reduce the number of memory reads/writes between GP…
We describe an efficient hierarchical method to compute attention in the Transformer architecture. The proposed attention mechanism exploits a matrix structure similar to the Hierarchical Matrix (H-Matrix) developed by the numerical analysis community, and has linear run time and memory complexity. We perform extensive experiments to show that the inductive bias embodied by our hierarchical attention is effective in capturing the hierarchical structure in the sequences typical for natural language and vision tasks. Our method is superior to alternative sub-quadratic proposals by over +6 points…
National Heart and Lung Institute, Imperial College London（帝国理工学院国家心肺研究所）；Cardiovascular Research Centre, Royal Brompton Hospital（皇家布朗普顿医院心血管研究中心）；School of Biomedical Engineering, Sun Yat-sen University（中山大学生物医学工程学院）
Fast MRI aims to reconstruct a high fidelity image from partially observed measurements. Exuberant development in fast MRI using deep learning has been witnessed recently. Meanwhile, novel deep learning paradigms, e.g., Transformer based models, are fast-growing in natural language processing and promptly developed for computer vision and medical image analysis due to their prominent performance. Nevertheless, due to the complexity of the Transformer, the application of fast MRI may not be straightforward. The main obstacle is the computational cost of the self-attention layer, which is the co…
FAST: A Synergistic Framework of Attention and State-space Models for Spatiotemporal Traffic Prediction
Columbia University（哥伦比亚大学）；San Francisco State University（旧金山州立大学）；University of California, Berkeley（加州大学伯克利分校）；New York University（纽约大学）；University of Pennsylvania（宾夕法尼亚大学）；University of California, San Diego（加州大学圣地亚哥分校）；University of Massachusetts Amherst（马萨诸塞大学阿姆赫斯特分校）；Carnegie Mellon University（卡内基梅隆大学）
Self-attention (SA) mechanisms can capture effectively global dependencies in deep neural networks, and have been applied to natural language processing and image processing successfully. However, SA modules for image reconstruction have high time and space complexity, which restrict their applications to higher-resolution images. In this paper, we refine the SA module in self-attention generative adversarial networks (SAGAN) via adapting a non-local operation, revising the connectivity among the units in SA module and re-implementing its computational pattern, such that its time and space com…
