This allows us to learn high-precision embeddings without using BigFloats, and enables us to store the resulting embeddings with fewer bits.

Neural Information Processing Systems http://nips.cc/

Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincar\'e ball model and the hyperboloid model as coordinate representations of the hyperbolic space, often neglecting the Klein model. Despite this, the Klein model offers its distinct advantages thanks to its straight-line geodesics, which facilitates the well-known Einstein midpoint construction, previously leveraged to accompany HNNs in other models. In this work, we introduce a framework for hyperbolic neural networks based on the Klein model. We provide detailed formulation for representing useful operations using the Klein model. We further study the Klein linear layer and prove that the "tangent space construction" of the scalar multiplication and parallel transport are exactly the Einstein scalar multiplication and the Einstein addition, analogous to the M\"obius operations used in the Poincar\'e ball model. We show numerically that the Klein HNN performs on par with the Poincar\'e ball model, providing a third option for HNN that works as a building block for more complicated architectures.

Graphs are data structures that extensively exist in real-world complex systems, varying from social networks [15, 62], protein interaction networks [52], recommender systems [9, 65, 64], knowledge graphs [56], to the financial transaction systems [40]. They form the basis of innumerable systems owing to their widespread utilization, allowing relational knowledge about interacting entities to be stored and accessible rapidly. Consequently, graph-related learning tasks gain increasing attention in machine learning and network science research. Many researchers have applied Graph Neural Networks (GNNs) for a variety of tasks, including node classification [23, 53, 59], link prediction [22, 71], and graph classification [61, 11] by embedding nodes in low-dimensional vector spaces, encoding topological and semantic information simultaneously. Many GNNs are built in Euclidean space in that it feature a vectorial structure, closed-form distance and inner-product formulae and is a natural extension of our intuitively appealing visual three-dimensional space [14]. Despite the effectiveness of Euclidean space for graph-related learning tasks, its ability to encode complex patterns is intrinsically limited by its polynomially expanding capacity. Although nonlinear techniques [3] assist in mitigating this issue, complex graph patterns may still need an embedding dimensionality that is computationally intractable. As revealed by recent research [4] many complex data show non-Euclidean underlying anatomy. For example, the tree-like structure extensively exists in many real-world networks, such as the hypernym structure in natural languages, the subordinate structure of entities in the knowledge graph, the organizational structure for financial fraud, and the power-law distribution in recommender systems.

The choice of geometric space for knowledge graph (KG) embeddings can have significant effects on the performance of KG completion tasks. The hyperbolic geometry has been shown to capture the hierarchical patterns due to its tree-like metrics, which addressed the limitations of the Euclidean embedding models. Recent explorations of the complex hyperbolic geometry further improved the hyperbolic embeddings for capturing a variety of hierarchical structures. However, the performance of the hyperbolic KG embedding models for non-transitive relations is still unpromising, while the complex hyperbolic embeddings do not deal with multi-relations. This paper aims to utilize the representation capacity of the complex hyperbolic geometry in multi-relational KG embeddings. To apply the geometric transformations which account for different relations and the attention mechanism in the complex hyperbolic space, we propose to use the fast Fourier transform (FFT) as the conversion between the real and complex hyperbolic space. Constructing the attention-based transformations in the complex space is very challenging, while the proposed Fourier transform-based complex hyperbolic approaches provide a simple and effective solution. Experimental results show that our methods outperform the baselines, including the Euclidean and the real hyperbolic embedding models.

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 reconsider a way to generalize the fundamental components of neural networks in a single hyperbolic geometry model, and propose novel methodologies to construct a multinomial logistic regression, fully-connected layers, convolutional layers, and attention mechanisms under a unified mathematical interpretation, without increasing the parameters. A series of experiments show the parameter efficiency of our methods compared to a conventional hyperbolic component, and stability and outperformance over their Euclidean counterparts.

This work presents a reformulation of the recently proposed Wasserstein autoencoder framework on a non-Euclidean manifold, the Poincar\'e ball model of the hyperbolic space. By assuming the latent space to be hyperbolic, we can use its intrinsic hierarchy to impose structure on the learned latent space representations. We demonstrate the model in the visual domain to analyze some of its properties and show competitive results on a graph link prediction task.

Well this was the reaction from quite a few people to whom I showed the results. The results seem good to me. In one of my previous post, Cross-lingual word embeddings- What they are?, I explained about word embeddings. They can be used in different tasks like information retrieval, sentiment analysis and myriad others. Similarly we can embed graphs, and have methods like node2vec, latent space embeddings which can help us in representing graphs and subsequently in community detection and link prediction.

Recently, Awasthi et al. introduced an SDP relaxation of the $k$-means problem in $\mathbb R^m$. In this work, we consider a random model for the data points in which $k$ balls of unit radius are deterministically distributed throughout $\mathbb R^m$, and then in each ball, $n$ points are drawn according to a common rotationally invariant probability distribution. For any fixed ball configuration and probability distribution, we prove that the SDP relaxation of the $k$-means problem exactly recovers these planted clusters with probability $1-e^{-\Omega(n)}$ provided the distance between any two of the ball centers is $>2+\epsilon$, where $\epsilon$ is an explicit function of the configuration of the ball centers, and can be arbitrarily small when $m$ is large.