Finding a low dimensional representation of hierarchical, structured data described by a network remains a challenging problem in the machine learning community. An emerging approach is embedding these networks into hyperbolic space because it can naturally represent a network's hierarchical structure. However, existing hyperbolic embedding approaches cannot deal with attributed networks, in which nodes are annotated with additional attributes. These attributes might provide additional proximity information to constrain the representations of the nodes, which is important to learn high quality hyperbolic embeddings. To fill this gap, we introduce HEAT (Hyperbolic Embedding of ATributed networks), the first method for embedding attributed networks to a hyperbolic space. HEAT consists of 1) a modified random walk algorithm to obtain training samples that capture both topological and attribute similarity; and 2) a learning algorithm for learning hyperboloid embeddings from the obtained training samples. We show that by leveraging node attributes, HEAT can outperform a state-of-the-art Hyperbolic embedding algorithm on several downstream tasks. As a general embedding method, HEAT opens the door to hyperbolic manifold learning on a wide range of attributed and unattributed networks.
Beyond the two-dimensional hyperbolic disk, an @math -dimensional Poincar 'e ball can give more degrees of freedom to the embedding and capture further dimensions of attractiveness than just popularity'' and similarity'' @cite_8 @cite_1 . By embedding general graphs to trees, @cite_0 were able to achieve state-of-the-art results by extending the work of @cite_19 .
@cite_0 @cite_19 @cite_1 @cite_8
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Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while 's recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable. We introduce a novel measure called e-four-pointscondition (e-4PC), which assigns a value e ∈ [0,1] to every metric space quantifying how close the metric is to a tree metric. Data-sets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based on this condition. We study embeddings of e-4PC metric spaces into trees and prove tight upper and lower bounds. Specifically, we show that there are constants c1 and c2 such that, (1) every metric (X,d) which satisfies the e-4PC can be embedded into a tree with distortion (1+e)c1log|X|, and (2) for every e ∈: [0,1] and any number of nodes, there is a metric space (X,d) satisfying the e-4PC that does not embed into a tree with distortion less than (1+e)c2log|X|. In addition, we prove a lower bound on approximate distance labelings of e-4PC metrics, and give tight bounds for tree embeddings with additive error guarantees. We introduce a novel projection-based visualization method for high-dimensional data sets by combining concepts from MDS and the geometry of the hyperbolic spaces. This approach hyperbolic multi-dimensional scaling (H-MDS) is a synthesis of two important concepts for explorative data analysis and visualization: (i) multi-dimensional scaling uses proximity or pair distance data to generate a low-dimensional, spatial presentation of the data; (ii) previous work on the "hyperbolic tree browser" demonstrated the extraordinary advantages for an interactive display of graph-like data in the two-dimensional hyperbolic space (H2).In the new approach, H-MDS maps proximity data directly into the H2. This removes the restriction to "quasihierarchical", graph-based data--a major limitation of (ii). Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data, this type of data can be considered the most general.We review important properties of the hyperbolic space and, in particular, the circular Poincare model of the H2. It enables effective human-computer interaction: by mouse dragging the "focus", the user can navigate in the data without loosing the context. In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.Several high-dimensional mapping examples including synthetic and real-world data are presented. Since high-dimensional data produce "ring"-shaped displays, we present methods to enhance the display by modulating the dissimilarity contrast. This is demonstrated for an application for unstructured text: i.e., by using multiple film critiques from news:rec.art.movies.reviews and www.imdb.com, each movie is placed within the H2--creating a "space of movies" for interactive exploration. This paper considers the problem of embedding trees into the hyperbolic plane. We show that any tree can be realized as the Delaunay graph of its embedded vertices. Particularly, a weighted tree can be embedded such that the weight on each edge is realized as the hyperbolic distance between its embedded vertices. Thus the embedding preserves the metric information of the tree along with its topology. The distance distortion between non adjacent vertices can be made arbitrarily small --- less than a (1+e) factor for any given e. Existing results on low distortion of embedding discrete metrics into trees carry over to hyperbolic metric through this result. The Delaunay character implies useful properties such as guaranteed greedy routing and realization as minimum spanning trees. We propose a novel projection based visualization method for high-dimensional datasets by combining concepts from MDS and the geometry of the hyperbolic spaces. Our approach Hyperbolic Multi-Dimensional Scaling (H-MDS) extends earlier work [7] using hyperbolic spaces for visualization of tree structures data ( "hyperbolic tree browser" ).By borrowing concepts from multi-dimensional scaling we map proximity data directly into the 2-dimensional hyperbolic space (H2). This removes the restriction to "quasihierarchical", graph-based data -- limiting previous work. Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data this type of data can be considered the most general.We used the circular Poincare model of the H2 which allows effective human-computer interaction: by moving the "focus" via mouse the user can navigate in the data without loosing the "context". In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.We present several high-dimensional mapping examples including synthetic and real world data and a successful application for unstructured text. By analyzing and integrating multiple film critiques from news:rec.art.movies.reviews and the internet movie database, each movie becomes placed within the H2. Here the idea is, that related films share more words in their reviews than unrelated. Their semantic proximity leads to a closer arrangement. The result is a kind of high-level content structured display allowing the user to explore the "space of movies". Beyond linear and kernel-based feature extraction, we propose in this paper the generalized feature extraction formulation based on the so-called graph embedding framework. Two novel correlation metric based algorithms are presented based on this formulation. correlation embedding analysis (CEA), which incorporates both correlational mapping and discriminating analysis, boosts the discriminating power by mapping data from a high-dimensional hypersphere onto another low-dimensional hypersphere and preserving the intrinsic neighbor relations with local graph modeling. correlational principal component analysis (CPCA) generalizes the conventional Principal Component Analysis (PCA) algorithm to the case with data distributed on a high-dimensional hypersphere. Their advantages stem from two facts: 1) tailored to normalized data, which are often the outputs from the data preprocessing step, and 2) directly designed with correlation metric, which shows to be generally better than Euclidean distance for classification purpose. Extensive comparisons with existing algorithms on visual classification experiments demonstrate the effectiveness of the proposed algorithms. We present the H3 layout technique for drawing large directed graphs as node-link diagrams in 3D hyperbolic space. We can lay out much larger structures than can be handled using traditional techniques for drawing general graphs because we assume a hierarchical nature of the data. We impose a hierarchy on the graph by using domain-specific knowledge to find an appropriate spanning tree. Links which are not part of the spanning tree do not influence the layout but can be selectively drawn by user request. The volume of hyperbolic 3-space increases exponentially, as opposed to the familiar geometric increase of euclidean 3-space. We exploit this exponential amount of room by computing the layout according to the hyperbolic metric. We optimize the cone tree layout algorithm for 3D hyperbolic space by placing children on a hemisphere around the cone mouth instead of on its perimeter. Hyperbolic navigation affords a Focus+Context view of the structure with minimal visual clutter. We have successfully laid out hierarchies of over 20,000 nodes. Our implementation accommodates navigation through graphs too large to be rendered interactively by allowing the user to explicitly prune or expand subtrees. In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to represent entities. Relations are modelled as rotations in the quaternion space. The advantages of the proposed approach are: (1) Latent inter-dependencies (between all components) are aptly captured with Hamilton product, encouraging a more compact interaction between entities and relations; (2) Quaternions enable expressive rotation in four-dimensional space and have more degree of freedom than rotation in complex plane; (3) The proposed framework is a generalization of ComplEx on hypercomplex space while offering better geometrical interpretations, concurrently satisfying the key desiderata of relational representation learning (i.e., modeling symmetry, anti-symmetry and inversion). Experimental results demonstrate that our method achieves state-of-the-art performance on four well-established knowledge graph completion benchmarks. Abstract This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum-length routing. More specifically, it is shown that in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is Θ(1), whereas in a Euclidean ball, the same proportion scales as Θ(1 R n−1). This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume2(B), whereas the same traffic load scales as volume1+1 n (B) in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic ... The Johnson--Lindenstrauss lemma shows that any n points in Euclidean space (i.e., ℝn with distances measured under the e 2 norm) may be mapped down to O((log n) e2) dimensions such that no pairwise distance is distorted by more than a (1 p e) factor. Determining whether such dimension reduction is possible in e 1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in e 1 . We give an explicit family of n points in e 1 such that any embedding with constant distortion D requires nΩ(1 D2) dimensions. This proves that there is no analog of the Johnson--Lindenstrauss lemma for e 1 ; in fact, embedding with any constant distortion requires nΩ(1) dimensions. Further, embedding the points into e 1 with (1pe) distortion requires nh−O(e log(1 e)) dimensions. Our proof establishes this lower bound for shortest path metrics of series-parallel graphs. We make extensive use of linear programming and duality in devising our bounds. We expect that the tools and techniques we develop will be useful for future investigations of embeddings into e 1 . δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points u,v,w,x, the two larger of the sums d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w) differ by at most 2δ. Given a finite set S of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of S with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for S with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs G=(V,E) with uniformly bounded degrees of vertices, the exact center of S can be computed in linear time O(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δlog2 n). This construction has an additive error comparable with that given by Gromov for n-point δ-hyperbolic spaces, but can be implemented in O(|E|) time (instead of O(n2)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity. We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature affects the heterogeneity of the degree distribution, while clustering is a function of temperature. We embed the internet into the hyperbolic plane and find a remarkable congruency between the embedding and our hyperbolic model. Besides proving our model realistic, this embedding may be used for routing with only local information, which holds significant promise for improving the performance of internet routing. δ-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points u,v,w,x, the two larger of the distance sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. In this paper, we study unweighted δ-hyperbolic graphs. Using the Layering Partition technique, we show that every n-vertex δ-hyperbolic graph with δ≥1 2 has an additive O(δlog n)-spanner with at most O(δn) edges and provide a simpler, in our opinion, and faster construction of distance approximating trees of δ-hyperbolic graphs with an additive error O(δlog n). The construction of our tree takes only linear time in the size of the input graph. As a consequence, we show that the family of n-vertex δ-hyperbolic graphs with δ≥1 2 admits a routing labeling scheme with O(δlog 2 n) bit labels, O(δlog n) additive stretch and O(log 2(4δ)) time routing protocol, and a distance labeling scheme with O(log 2 n) bit labels, O(δlog n) additive error and constant time distance decoder. We consider the problem of covering and packing subsets ofΔ-hyperbolic metric spaces and graphs by balls.These spaces, defined via a combinatorial Gromov condition, haverecently become of interest in several domains of computer science.Specifically, given a subset Sof aΔ-hyperbolic graph Gand a positive numberR, let Δ(S,R) be theminimum number of balls of radius Rcovering S.It is known that computing Δ(S,R)or approximating this number within a constant factor is hard evenfor 2-hyperbolic graphs. In this paper, using a primal-dualapproach, we show how to construct in polynomial time a covering ofSwith at most Δ(S,R)balls of (slightly larger) radius R+ Δ.This result is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extendedto some other set families derived from balls. The coveringalgorithm is used to design better approximation algorithms for theaugmentation problem with diameter constraints and for thek-center problem in Δ-hyperbolicgraphs. Low-dimensional embedding based on non-metric data (e.g., non-metric multidimensional scaling) is a problem that arises in many applications, especially those involving human subjects. This paper investigates the problem of learning an embedding of n objects into d-dimensional Euclidean space that is consistent with pairwise comparisons of the type “object a is closer to object b than c.” While there are O(n3) such comparisons, experimental studies suggest that relatively few are necessary to uniquely determine the embedding up to the constraints imposed by all possible pairwise comparisons (i.e., the problem is typically over-constrained). This paper is concerned with quantifying the minimum number of pairwise comparisons necessary to uniquely determine an embedding up to all possible comparisons. The comparison constraints stipulate that, with respect to each object, the other objects are ranked relative to their proximity. We prove that at least Q(dn log n) pairwise comparisons are needed to determine the embedding of all n objects. The lower bounds cannot be achieved by using randomly chosen pairwise comparisons. We propose an algorithm that exploits the low-dimensional geometry in order to accurately embed objects based on relatively small number of sequentially selected pairwise comparisons and demonstrate its performance with experiments. We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into ℝ d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1 12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1 (22d-10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala), For d > 3, we obtain a stronger inapproximability result by a different reduction: assuming P≠NP, no polynomial-time algorithm can distinguish between spaces embeddable in ℝ d with constant distortion from spaces requiring distortion at least n c d , for a constant c > 0. The exponent c d has the correct order of magnitude, since every n-point metric space can be embedded in ℝ d with distortion O(n 2 d log 3 2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in ℝ 2 , while all not too large subspaces of it embed almost isometrically. We consider the problem of embedding general metrics into trees. We give the first non-trivial approximation algorithm for minimizing the multiplicative distortion. Our algorithm produces an embedding with distortion (c log n)O(√log Δ), where c is the optimal distortion, and Δ is the spread of the metric (i.e. the ratio of the diameter over the minimum distance). We give an improved O(1)-approximation algorithm for the case where the input is the shortest path metric over an unweighted graph. Moreover, we show that by composing our approximation algorithm for embedding general metrics into trees, with the approximation algorithm of [BCIS05] for embedding trees into the line, we obtain an improved approximation algorithm for embedding general metrics into the line. We also provide almost tight bounds for the relation between embedding into trees and embedding into spanning subtrees. We show that for any unweighted graph G, the ratio of the distortion required to embed G into a spanning subtree, over the distortion of an optimal tree embedding of G, is at most O(log n). We complement this bound by exhibiting a family of graphs for which the ratio is Ω(log n log log n). Physicists recently observed that realistic complex networks emerge as discrete samples from a continuous hyperbolic geometry enclosed in a circle: the radius represents the node centrality and the angular displacement between two nodes resembles their topological proximity. The hyperbolic circle aims to become a universal space of representation and analysis of many real networks. Yet, inferring the angular coordinates to map a real network back to its latent geometry remains a challenging inverse problem. Here, we show that intelligent machines for unsupervised recognition and visualization of similarities in big data can also infer the network angular coordinates of the hyperbolic model according to a geometrical organization that we term “angular coalescence.” Based on this phenomenon, we propose a class of algorithms that offers fast and accurate “coalescent embedding” in the hyperbolic circle even for large networks. This computational solution to an inverse problem in physics of complex systems favors the application of network latent geometry techniques in disciplines dealing with big network data analysis including biology, medicine, and social science. Mapping complex networks to underlying geometric spaces can help understand the structure of networked systems. Here the authors propose a class of machine learning algorithms for efficient embedding of large real networks to the hyperbolic space, with potential impact on big network data analysis. We study the problem of embedding weighted graphs of pathwidth k into lp spaces. Our main result is an O(kmin 1p,12 )-distortion embedding. For p=1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p >1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p>2 our embedding also implies improved distortion on bounded treewidth graphs (O((klogn)1p)). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor. We consider monotone embeddings of a finite metric space into low-dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l2n, while (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension Ω(n) (Lemma 3).It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on sphericity of graphs, which suggest one example of such a metric space--that is defined by a complete bipartite graph. We prove that an δn-regular graph of order n, with bounded diameter has sphericity Ω(n (λ2+1)), where λ2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0<δ≤½ is constant (Theorem 4). We also show that while random graphs have linear sphericity, there are quais-random graphs of logarithmic sphericity (Lemma 7).For the above bound to be linear, λ2 must be constant. We show that if the second eigenvalue of an n 2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2) entries (Theorem 5). Furthermore, for any 0<δ<½, and λ2, there are only finitely many δn-regular graphs with second eigenvalue at most λ2 (Corollary 4).