HyperSoRec: Exploiting Hyperbolic User and Item Representations with Multiple Aspects for Social-aware Recommendation

Social recommendation has achieved great success in many domains including e-commerce and location-based social networks. Existing methods usually explore the user-item interactions or user-user connections to predict users’ preference behaviors. However, they usually learn both user and item representations in Euclidean space, which has large limitations for exploring the latent hierarchical property in the data. In this article, we study a novel problem of hyperbolic social recommendation, where we aim to learn the compact but strong representations for both users and items. Meanwhile, this work also addresses two critical domain-issues, which are under-explored. First, users often make trade-offs with multiple underlying aspect factors to make decisions during their interactions with items. Second, users generally build connections with others in terms of different aspects, which produces different influences with aspects in social network. To this end, we propose a novel graph neural network (GNN) framework with multiple aspect learning, namely, HyperSoRec. Specifically, we first embed all users, items, and aspects into hyperbolic space with superior representations to ensure their hierarchical properties. Then, we adapt a GNN with novel multi-aspect message-passing-receiving mechanism to capture different influences among users. Next, to characterize the multi-aspect interactions of users on items, we propose an adaptive hyperbolic metric learning method by introducing learnable interactive relations among different aspects. Finally, we utilize the hyperbolic translational distance to measure the plausibility in each user-item pair for recommendation. Experimental results on two public datasets clearly demonstrate that our HyperSoRec not only achieves significant improvement for recommendation performance but also shows better representation ability in hyperbolic space with strong robustness and reliability.

Visualization of Escher-like Spiral Patterns in Hyperbolic Space

Spirals, tilings, and hyperbolic geometry are important mathematical topics with outstanding aesthetic elements. Nonetheless, research on their aesthetic visualization is extremely limited. In this paper, we give a simple method for creating Escher-like hyperbolic spiral patterns. To this end, we first present a fast algorithm to construct Euclidean spiral tilings with cyclic symmetry. Then, based on a one-to-one mapping between Euclidean and hyperbolic spaces, we establish two simple approaches for constructing spiral tilings in hyperbolic models. Finally, we use wallpaper templates to render such tilings, which results in the desired Escher-like hyperbolic spiral patterns. The method proposed is able to generate a great variety of visually appealing patterns.

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

<p style='text-indent:20px;'>This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [<xref ref-type="bibr" rid="b26">26</xref>] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.</p>

Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

Some type of semisymmetry on two classes of almost Kenmotsu manifolds

The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a (k, µ)-almost Kenmotsu manifold satisfying the curvature condition Q · R = 0 is locally isometric to the hyperbolic space ℍ2 n +1(−1). Also in (k, µ)-almost Kenmotsu manifolds the following conditions: (1) local symmetry (∇R = 0), (2) semisymmetry (R·R = 0), (3) Q(S, R) = 0, (4) R·R = Q(S, R), (5) locally isometric to the hyperbolic space ℍ2 n +1(−1) are equivalent. Further, it is proved that a (k, µ)′ -almost Kenmotsu manifold satisfying Q · R = 0 is locally isometric to ℍ n +1(−4) × ℝ n and a (k, µ)′ -almost Kenmotsu manifold satisfying any one of the curvature conditions Q(S, R) = 0 or R · R = Q(S, R) is either an Einstein manifold or locally isometric to ℍ n +1(−4) × ℝ n . Finally, an illustrative example is presented.