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Hyperbolic Deep Neural Networks: a Survey JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 Hyperbolic Deep Neural Networks: A Survey Wei Peng, Tuomas Varanka, Abdelrahman Mostafa, Henglin Shi, Guoying Zhao* Senior Member, IEEE Abstract—Recently, there has been a rising surge of momentum for deep representation learning in hyperbolic spaces due to their high 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 can potentially lead to drastically more compact models with much more physical interpretability than its counterpart in the Euclidean space. To stimulate future research, this paper presents a coherent and a comprehensive review of the literature around the neural components in the construction of hyperbolic deep neural networks, as well as the generalization of the leading deep approaches to the hyperbolic space. It also presents current applications around various machine learning tasks on several publicly available datasets, together with insightful observations and identifying open questions and promising future directions. Index Terms—Deep Neural Networks, Hyperbolic Geometry, Neural Networks on Riemannian Manifold, Hyperbolic Neural Networks, Poincare´ Model, Lorentz Model, Negative Curvature Geometry. F 1 INTRODUCTION curvature, and it is analogous to a high-dimensional sphere with Earning semantically rich representations of data such as text constant positive curvature. L or images is a central point of interest in current machine In many domains, data is with a tree-like structure or can learning. Recently, deep neural networks [1] have showcased a be represented hierarchically [14]. For instance, social networks, great capability in extracting feature representations and have human skeletons, sentences in natural language, and evolution- dominated many research fields, such as image classification [2], ary relationships between biological entities in phylogenetics. [3], machine translation tasks [4], and playing video games [5]. As also mentioned by [15], a wide variety of real-world data Theoretically, deep neural networks with millions of parameters encompasses some types of latent hierarchical structures [16], have great potential to fit any complex functions, especially when [17], [18]. Besides, from the perspective of cognitive science, they are equipped with many advanced optimization libraries [6], it is widely accepted that human beings use hierarchy to or- [7], [8]. At the core of deep neural networks lies the expectation ganise object categories [19], [20], [21]. As a result, there is to find the optimal representations (task-related), of which the un- always a passion to model the data hierarchically. Explicitly derlying manifold assumption is that the intrinsic dimensionality incorporating hierarchical structure in probabilistic models has of the input data is much lower compared to the input feature unsurprisingly been a long-running research topic. Earlier work space dimension. In most of the current deep learning applications, in this direction tended towards using trees as data structures the representation learning is conducted in the Euclidean space, to represent hierarchies. Recently, hyperbolic spaces have been which is reasonable since the Euclidean space is the natural proposed as an alternative continuous approach to learn hierar- generalization of our intuition-friendly, visual three-dimensional chical representations from textual and graph-structured data. The space. However, recent research has shown that many types of negative-curvature of the hyperbolic space results in very different complex data exhibit a highly non-Euclidean latent anatomy [9]. geometric properties, which makes it widely employed in many Besides, it appears in several applications that the dissimilarity settings. In the hyperbolic space, circle circumference (2 sinh A) measures constructed by experts tend to have non-Euclidean and disc area (2c¹ cosh A−1º) grow exponentially with radius A, as behavior [10]. In such cases, the Euclidean space does not provide opposed to the Euclidean space where they only grow linearly and arXiv:2101.04562v3 [cs.LG] 17 Feb 2021 the most powerful or meaningful geometrical representations. quadratically. The exponential growth of the Poincare´ surface area Works like [11] even suggest that data representations in most with respect to its radius is analogous to the exponential growth of machine learning applications lie on a smooth manifold, and the the number of leaves in a tree with respect to its depth, rather than smooth manifold is Riemannian, which is equipped with a positive polynomially as in the Euclidean case [22], [23]. On the contrary, definite metric on each tangent space, i.e., every non-vanishing Euclidean space, as shown in Bourgain’s theorem [24], is unable tangent vector has a positive squared norm. Due to the positive to obtain comparably low distortion for tree data, even when definiteness of the metric, the negative of the (Riemannian) gra- using an unbounded number of dimensions. Furthermore, the dient is a descent direction that can be exploited to iteratively hyperbolic spaces are smooth, enabling the use of deep learning minimize some objective functions. Therefore current research approaches which rely on differentiability. Therefore, hyperbolic is increasingly attracted by the idea of building neural networks spaces have recently gained momentum in the context of deep in a Riemannian space, such as the hyperbolic space [12], [13], neural networks to model embedded data into the space that which is a Riemannian manifold with constant negative (sectional) exhibits certain desirable geometric characteristics. To summarize, there are several potential advantages of utilizing hyperbolic deep • W. Peng, T. Varanka, A. Mostafa, H. Shi, and G. Zhao are with the Center neural networks to represent data: for Machine Vision and Signal Analysis, University of Oulu, Oulu, Finland. E-mail: firstname.lastname@oulu.fi • A better generalization capability of the model, with less Manuscript received April 19, 2020; revised August 26, 2020. overfitting, computational complexity, and requirement of JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2 training data. Euclidean Geometry define the basic rules governing the creation • Reduction in the number of model parameters and embed- and extension of geometric figures with a ruler and compass. They ding dimensions. are as follows: • A low distortion embedding, which preserves the local and • A straight line segment can be drawn joining any two geometric information. points. • A better model understanding and interpretation. • Any straight line segment may be extended to any finite Hyperbolic deep neural networks have a great potential in both length. academia and industry. For instance, in academia, the negatively • A circle may be described with any given point as its center curved spaces have already gained much attention for embedding and any distance as its radius. and representing relationships between objects that are organized • All right angles are congruent. hierarchically [23], [25]. In industry, there are already recom- • If two lines are drawn which intersect a third in such a mender systems based on hyperbolic models, which are scaled way that the sum of the inner angles on one side is less to millions of users [26]. Also, we can find work for learning than two right angles, then the two lines inevitably must drug hierarchy [27]. Although constructing neural networks on intersect each other on that side if extended far enough. hyperbolic space gained great attention, as far as we know, there This postulate is equivalent to what is known as the parallel is currently no survey paper in this field. This article makes the postulate. first attempt and aims to provide a comprehensive review of the To better help understand the last postulate Fig 1 shows equiv- literature around hyperbolic deep neural networks for machine alent statements pictorially. Here we take a look at the parallel learning tasks. Our goals are to 1) provide concise context and postulate, which is shown in Fig. 1 (b) and its definition is that explanation to enable the reader to become familiar with the basics given any straight line and a point not on it, “there exists one and of hyperbolic geometry, 2) review the current literature related to only one straight line which passes through that point and never algorithms and applications, and 3) identify open questions and intersects the first line, no matter how far they are extended”. This promising future directions. We hope this article could be a tutorial statement is equivalent to the fifth postulate and it was the source for this topic, as well as a formal theoretical support for future of much annoyance, and for at least a thousand years, geometers research. were troubled by the disparate complexity of the fifth postulate. The article is organized as follows. In Section 2, we introduce As a result, many mathematicians over the centuries have tried the fundamental concepts about hyperbolic geometry, making the to prove the results of the Elements without using the Parallel paper self-contained. Section 3 introduces the generalization of Postulate, but to no avail. However, in the past two centuries, important neural network components from the Euclidean space assorted non-Euclidean geometries have been derived based on to the hyperbolic space. We then review the constructions
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