National Special Education Resource Center for Children with Autism, Zhejiang Normal University（浙江师范大学国家孤独症儿童特殊教育资源中心）；Institute of Cyber-Systems and Control, Zhejiang University（浙江大学 cyber 系统与控制研究所）；School of Computer Science and Technology, Zhejiang Normal University（浙江师范大学计算机科学与技术学院）
Jun Chen, Jingyang Xiang, Tianxin Huang, Xiangrui Zhao, Yong Liu
发表信息：
IEEE Transactions on Neural Networks and Learning Systems, 2024
Binary Neural Network (BNN) converts full-precision weights and activations into their extreme 1-bit counterparts, making it particularly suitable for deployment on lightweight mobile devices. While binary neural networks are typically formulated as a constrained optimization problem and optimized in the binarized space, general neural networks are formulated as an unconstrained optimization problem and optimized in the continuous space. This paper introduces the Hyperbolic Binary Neural Network (HBNN) by leveraging the framework of hyperbolic geometry to optimize the constrained problem. Specifically, we transform the constrained problem in hyperbolic space into an unconstrained one in Euclidean space using the Riemannian exponential map. On the other hand, we also propose the Exponential Parametrization Cluster (EPC) method, which, compared to the Riemannian exponential map, shrinks the segment domain based on a diffeomorphism. This approach increases the probability of weight flips, thereby maximizing the information gain in BNNs. Experimental results on CIFAR10, CIFAR100, and ImageNet classification datasets with VGGsmall, ResNet18, and ResNet34 models illustrate the superior performance of our HBNN over state-of-the-art methods.
We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments.
Redwood Center for Theoretical Neuroscience（红木理论神经科学中心）；University of California, Berkeley（加州大学伯克利分校）
Recent research has shown that one can train a neural network with binary weights and activations at train time by augmenting the weights with a high-precision continuous latent variable that accumulates small changes from stochastic gradient descent. However, there is a dearth of theoretical analysis to explain why we can effectively capture the features in our data with binary weights and activations. Our main result is that the neural networks with binary weights and activations trained using the method of Courbariaux, Hubara et al. (2016) work because of the high-dimensional geometry of bi…
Recently, there has been a rising surge of momentum for deep representation learning in hyperbolic spaces due to theirhigh capacity of modeling data like knowledge graphs or synonym hierarchies, possessing hierarchical structure. We refer to the model as hyperbolic deep neural network in this paper. Such a hyperbolic neural architecture potentially leads to drastically compact model withmuch more physical interpretability than its counterpart in Euclidean space. To stimulate future research, this paper presents acoherent and comprehensive review of the literature around the neural components i…
UC Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）；Facebook AI Research（脸书人工智能研究院）
Hyperbolic space can naturally embed hierarchies, unlike Euclidean space. Hyperbolic Neural Networks (HNNs) exploit such representational power by lifting Euclidean features into hyperbolic space for classification, outperforming Euclidean neural networks (ENNs) on datasets with known semantic hierarchies. However, HNNs underperform ENNs on standard benchmarks without clear hierarchies, greatly restricting HNNs' applicability in practice. Our key insight is that HNNs' poorer general classification performance results from vanishing gradients during backpropagation, caused by their hybrid archi…
Many graph neural networks have been developed to learn graph representations in either Euclidean or hyperbolic space, with all nodes' representations embedded in a single space. However, a graph can have hyperbolic and Euclidean geometries at different regions of the graph. Thus, it is sub-optimal to indifferently embed an entire graph into a single space. In this paper, we explore and analyze two notions of local hyperbolicity, describing the underlying local geometry: geometric (Gromov) and model-based, to determine the preferred space of embedding for each node. The two hyperbolicities' di…
Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces. As a result, we derive hyperbo…
Tsinghua University（清华大学）；Tencent Inc（腾讯公司）；Beijing Academy of Artificial Intelligence（北京智源人工智能研究院）
Hyperbolic neural networks have shown great potential for modeling complex data. However, existing hyperbolic networks are not completely hyperbolic, as they encode features in a hyperbolic space yet formalize most of their operations in the tangent space (a Euclidean subspace) at the origin of the hyperbolic space. This hybrid method greatly limits the modeling ability of networks. In this paper, we propose a fully hyperbolic framework to build hyperbolic networks based on the Lorentz model by adapting the Lorentz transformations (including boost and rotation) to formalize essential operation…
The University of British Columbia（不列颠哥伦比亚大学）；Xtract AI；Computational Geosciences Inc.（计算地球科学公司）
Convolutional Neural Networks (CNN) have recently seen tremendous success in various computer vision tasks. However, their application to problems with high dimensional input and output, such as high-resolution image and video segmentation or 3D medical imaging, has been limited by various factors. Primarily, in the training stage, it is necessary to store network activations for back propagation. In these settings, the memory requirements associated with storing activations can exceed what is feasible with current hardware, especially for problems in 3D. Motivated by the propagation of signal…
Learning from graph-structured data is an important task in machine learning and artificial intelligence, for which Graph Neural Networks (GNNs) have shown great promise. Motivated by recent advances in geometric representation learning, we propose a novel GNN architecture for learning representations on Riemannian manifolds with differentiable exponential and logarithmic maps. We develop a scalable algorithm for modeling the structural properties of graphs, comparing Euclidean and hyperbolic geometry. In our experiments, we show that hyperbolic GNNs can lead to substantial improvements on var…
RIKEN AIP（理化学研究所人工智能项目）；The University of Tokyo（东京大学）
Hyperbolic spaces, which have the capacity to embed tree structures without distortion owing to their exponential volume growth, have recently been applied to machine learning to better capture the hierarchical nature of data. In this study, we generalize the fundamental components of neural networks in a single hyperbolic geometry model, namely, the Poincaré ball model. This novel methodology constructs a multinomial logistic regression, fully-connected layers, convolutional layers, and attention mechanisms under a unified mathematical interpretation, without increasing the parameters. Experi…
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unc…
Hyperbolic networks have shown prominent improvements over their Euclidean counterparts in several areas involving hierarchical datasets in various domains such as computer vision, graph analysis, and natural language processing. However, their adoption in practice remains restricted due to (i) non-scalability on accelerated deep learning hardware, (ii) vanishing gradients due to the closure of hyperbolic space, and (iii) information loss due to frequent mapping between local tangent space and fully hyperbolic space. To tackle these issues, we propose the approximation of hyperbolic operators …
Beijing University of Posts and Telecommunications（北京邮电大学）；Florida State University（佛罗里达州立大学）；Beihang University（北京航空航天大学）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
Learning representations for graphs plays a critical role in a wide spectrum of downstream applications. In this paper, we summarize the limitations of the prior works in three folds: representation space, modeling dynamics and modeling uncertainty. To bridge this gap, we propose to learn dynamic graph representation in hyperbolic space, for the first time, which aims to infer stochastic node representations. Working with hyperbolic space, we present a novel Hyperbolic Variational Graph Neural Network, referred to as HVGNN. In particular, to model the dynamics, we introduce a Temporal GNN (TGN…
Beihang University（北京航空航天大学）；Macquarie University（麦考瑞大学）；Central South University（中南大学）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
Graph Neural Networks (GNNs) have been widely studied in various graph data mining tasks. Most existingGNNs embed graph data into Euclidean space and thus are less effective to capture the ubiquitous hierarchical structures in real-world networks. Hyperbolic Graph Neural Networks(HGNNs) extend GNNs to hyperbolic space and thus are more effective to capture the hierarchical structures of graphs in node representation learning. In hyperbolic geometry, the graph hierarchical structure can be reflected by the curvatures of the hyperbolic space, and different curvatures can model different hierarch…
Neural networks and particularly Deep learning have been comparatively little studied from the theoretical point of view. Conversely, Mathematical Morphology is a discipline with solid theoretical foundations. We combine these domains to propose a new type of neural architecture that is theoretically more explainable. We introduce a Binary Morphological Neural Network (BiMoNN) built upon the convolutional neural network. We design it for learning morphological networks with binary inputs and outputs. We demonstrate an equivalence between BiMoNNs and morphological operators that we can use to b…
In the last ten years, Convolutional Neural Networks (CNNs) have formed the basis of deep-learning architectures for most computer vision tasks. However, they are not necessarily optimal. For example, mathematical morphology is known to be better suited to deal with binary images. In this work, we create a morphological neural network that handles binary inputs and outputs. We propose their construction inspired by CNNs to formulate layers adapted to such images by replacing convolutions with erosions and dilations. We give explainable theoretical results on whether or not the resulting learne…
Neural embeddings have been used with great success in Natural Language Processing (NLP). They provide compact representations that encapsulate word similarity and attain state-of-the-art performance in a range of linguistic tasks. The success of neural embeddings has prompted significant amounts of research into applications in domains other than language. One such domain is graph-structured data, where embeddings of vertices can be learned that encapsulate vertex similarity and improve performance on tasks including edge prediction and vertex labelling. For both NLP and graph based tasks, em…
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Binarization of neural network models is considered as one of the promising methods to deploy deep neural network models on resource-constrained environments such as mobile devices. However, Binary Neural Networks (BNNs) tend to suffer from severe accuracy degradation compared to the full-precision counterpart model. Several techniques were proposed to improve the accuracy of BNNs. One of the approaches is to balance the distribution of binary activations so that the amount of information in the binary activations becomes maximum. Based on extensive analysis, in stark contrast to previous work…
Hyperbolic Graph Neural Networks Under the Microscope: The Role of Geometry-Task Alignment
University of Rome Tor Vergata（罗马大学Tor Vergata）；University of Basel（巴塞尔大学）；University of Rome Tor Vergata Harvard Medical School（罗马大学Tor Vergata哈佛医学院）；Bocconi University Bocconi Institute for Data Science and Analytics（博科尼大学博科尼数据科学与分析研究所）；UiT The Arctic University of Norway（UiT北极大学）
Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks ap…
With the recent advance of geometric deep learning, neural networks have been extensively used for data in non-Euclidean domains. In particular, hyperbolic neural networks have proved successful in processing hierarchical information of data. However, many hyperbolic neural networks are numerically unstable during training, which precludes using complex architectures. This crucial problem makes it difficult to build hyperbolic generative models for real and complex data. In this work, we propose a hyperbolic generative network in which we design novel architecture and layers to improve stabili…
Department of Mathematics, Purdue University（普渡大学数学系）；School of Mechanical Engineering, Purdue University（普渡大学机械工程学院）
We introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume sc…
University of Florida（佛罗里达大学）；National Taiwan University（国立台湾大学）
Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network whic…
HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws
Laboratory of Computer Science (LIX), École Polytechnique（计算机科学实验室（LIX），巴黎综合理工学院）；Department of Electrical Engineering and Computer Sciences, University of California, Berkeley（电气工程与计算机科学系，加利福尼亚大学伯克利分校）
Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical met…
Hyperbolic Graph Neural Networks: A Review of Methods and Applications
Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Huawei Technologies Co., Ltd.（华为技术有限公司）；The Chinese University of Hong Kong（香港中文大学）；Zhejiang University（浙江大学）
Explaining Hyperbolic Neural Networks via Geometry-Aware Relevance Propagation
BIFOLD（柏林智能与数据科学基础研究所）；Technische Universität Berlin（柏林工业大学）；Nanyang Technological University（南洋理工大学）；University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；RIKEN Center for AIP（理化学研究所先进智能研究中心）
Hyperbolic Hypergraph Neural Networks for Multi-Relational Knowledge Hypergraph Representation
National Engineering Research Center for Big Data Technology and System（国家大数据技术与系统工程研究中心）；School of Computer Science and Technology（计算机科学与技术学院）；Huazhong University of Science and Technology（华中科技大学）
Curvature Learning for Generalization of Hyperbolic Neural Networks
Beijing Laboratory of Intelligent Information Technology（北京智能信息科技实验室）；School of Computer Science, Beijing Institute of Technology (BIT)（北京理工大学计算机学院）；Guangdong Laboratory of Machine Perception and Intelligent Computing（广东机器感知与智能计算实验室）；Shenzhen MSU-BIT University（深圳MSU-BIT大学）；Department of Electrical and Computer Systems Eng.（电气与计算机系统工程系）；Data61
Universidad Técnica Federico Santa María（费德里科·圣玛丽亚技术大学）
Recently, Hyperbolic Spaces in the context of Non-Euclidean Deep Learning have gained popularity because of their ability to represent hierarchical data. We propose that it is possible to take advantage of the hierarchical characteristic present in the images by using hyperbolic neural networks in a GAN architecture. In this study, different configurations using fully connected hyperbolic layers in the GAN, CGAN, and WGAN are tested, in what we call the HGAN, HCGAN, and HWGAN, respectively. The results are measured using the Inception Score (IS) and the Fréchet Inception Distance (FID) on the …
