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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 Poincare 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. Experiments show the superior parameter efficiency of our methods compared to conventional hyperbolic components, and stability and outperformance over their Euclidean counterparts.
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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 unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.
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 in the construction of hyperbolic deep neuralnetworks, as well as the generalization of the leading deep approaches to the Hyperbolic space. It also presents current applicationsaround various machine learning tasks on several publicly available datasets, together with insightful observations and identifying openquestions and promising future directions.
Modern neural networks often contain significantly more parameters than the size of their training data. We show that this excess capacity provides an opportunity for embedding secret machine learning models within a trained neural network. Our novel framework hides the existence of a secret neural network with arbitrary desired functionality within a carrier network. We prove theoretically that the secret networks detection is computationally infeasible and demonstrate empirically that the carrier network does not compromise the secret networks disguise. Our paper introduces a previously unknown steganographic technique that can be exploited by adversaries if left unchecked.
Non-Euclidean geometry with constant negative curvature, i.e., hyperbolic space, has attracted sustained attention in the community of machine learning. Hyperbolic space, owing to its ability to embed hierarchical structures continuously with low distortion, has been applied for learning data with tree-like structures. Hyperbolic Neural Networks (HNNs) that operate directly in hyperbolic space have also been proposed recently to further exploit the potential of hyperbolic representations. While HNNs have achieved better performance than Euclidean neural networks (ENNs) on datasets with implicit hierarchical structure, they still perform poorly on standard classification benchmarks such as CIFAR and ImageNet. The traditional wisdom is that it is critical for the data to respect the hyperbolic geometry when applying HNNs. In this paper, we first conduct an empirical study showing that the inferior performance of HNNs on standard recognition datasets can be attributed to the notorious vanishing gradient problem. We further discovered that this problem stems from the hybrid architecture of HNNs. Our analysis leads to a simple yet effective solution called Feature Clipping, which regularizes the hyperbolic embedding whenever its norm exceeding a given threshold. Our thorough experiments show that the proposed method can successfully avoid the vanishing gradient problem when training HNNs with backpropagation. The improved HNNs are able to achieve comparable performance with ENNs on standard image recognition datasets including MNIST, CIFAR10, CIFAR100 and ImageNet, while demonstrating more adversarial robustness and stronger out-of-distribution detection capability.
In this paper, we present a hypergraph neural networks (HGNN) framework for data representation learning, which can encode high-order data correlation in a hypergraph structure. Confronting the challenges of learning representation for complex data in real practice, we propose to incorporate such data structure in a hypergraph, which is more flexible on data modeling, especially when dealing with complex data. In this method, a hyperedge convolution operation is designed to handle the data correlation during representation learning. In this way, traditional hypergraph learning procedure can be conducted using hyperedge convolution operations efficiently. HGNN is able to learn the hidden layer representation considering the high-order data structure, which is a general framework considering the complex data correlations. We have conducted experiments on citation network classification and visual object recognition tasks and compared HGNN with graph convolutional networks and other traditional methods. Experimental results demonstrate that the proposed HGNN method outperforms recent state-of-the-art methods. We can also reveal from the results that the proposed HGNN is superior when dealing with multi-modal data compared with existing methods.
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