Related papers: Hyperbolic tiling neighborhoods in O(1) time

Given $n$ points in the plane, we propose algorithms to compile connected crossing-free geometric graphs into directed acyclic graphs (DAGs). The DAGs allow efficient counting, enumeration, random sampling, and optimization. Our algorithms…

Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform…

Graph-based nearest neighbor search methods have seen a surge of popularity in recent years, offering state-of-the-art performance across a wide variety of applications. Central to these methods is the task of constructing a sparse…

Using a hyperbolic complex plane, we study the realization of the underlying hyperbolic symmetry as an internal symmetry that enables the unification of scalar fields of cosmological and particle physics interest. Such an unification is…

Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored…

In this paper, we complete the construction of paper arXiv:cs.CG/0701096v2. Together with the proof contained in arXiv:cs.CG/0701096v2, this paper definitely proves that the general problem of tiling the hyperbolic plane with {\it \`a la}…

A new class of parallel algorithms is introduced that can achieve a complexity of O(n^3/2) with respect to the interprocessor communication, in the exact computation of systems with pairwise mutual interactions of all elements. Hitherto,…

We study the problem of clustering networks whose nodes have imputed or physical positions in a single dimension, for example prestige hierarchies or the similarity dimension of hyperbolic embeddings. Existing algorithms, such as the…

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Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated…

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Discrete geometries in hyperbolic space are of longstanding interest in pure mathematics and have come to recent attention in holography, quantum information, and condensed matter physics. Working at a purely geometric level, we describe…

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Temporal link prediction, aiming to predict future edges between paired nodes in a dynamic graph, is of vital importance in diverse applications. However, existing methods are mainly built upon uniform Euclidean space, which has been found…

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We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of…

Scene graph representations enable structured visual understanding by modeling objects and their relationships, and have been widely used for multiview and 3D scene reasoning. Existing methods such as MSG learn scene graph embeddings in…

In large-scale recommender systems, the user-item networks are generally scale-free or expand exponentially. The latent features (also known as embeddings) used to describe the user and item are determined by how well the embedding space…

Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…

Tree embedding has been a fundamental method in algorithm design with wide applications. We focus on the efficiency of building tree embedding in various computational settings under high-dimensional Euclidean $\mathbb{R}^d$. We devise a…