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Hierarchical relations are prevalent and indispensable for organizing human knowledge captured by a knowledge graph (KG). The key property of hierarchical relations is that they induce a partial ordering over the entities, which needs to be modeled in order to allow for hierarchical reasoning. However, current KG embeddings can model only a single global hierarchy (single global partial ordering) and fail to model multiple heterogeneous hierarchies that exist in a single KG. Here we present ConE (Cone Embedding), a KG embedding model that is able to simultaneously model multiple hierarchical as well as non-hierarchical relations in a knowledge graph. ConE embeds entities into hyperbolic cones and models relations as transformations between the cones. In particular, ConE uses cone containment constraints in different subspaces of the hyperbolic embedding space to capture multiple heterogeneous hierarchies. Experiments on standard knowledge graph benchmarks show that ConE obtains state-of-the-art performance on hierarchical reasoning tasks as well as knowledge graph completion task on hierarchical graphs. In particular, our approach yields new state-of-the-art Hits@1 of 45.3% on WN18RR and 16.1% on DDB14 (0.231 MRR). As for hierarchical reasoning task, our approach outperforms previous best results by an average of 20% across the three datasets. 

ABSTRACT We report on detailed multiwavelength observations and analysis of the very bright and long GRB 210619B, detected by the Atmosphere-Space Interactions Monitor installed on the International Space Station and the Gamma-ray Burst Monitor (GBM) on-board the Fermi mission. Our main goal is to understand the radiation mechanisms and jet composition of GRB 210619B. With a measured redshift of z = 1.937, we find that GRB 210619B falls within the 10 most luminous bursts observed by Fermi so far. The energy-resolved prompt emission light curve of GRB 210619B exhibits an extremely bright hard emission pulse followed by softer/longer emission pulses. The low-energy photon index (αpt) values obtained using the time-resolved spectral analysis of the burst suggest a transition between the thermal (during harder pulse) to non-thermal (during softer pulse) outflow. We examine the correlation between spectral parameters and find that both peak energy and αpt exhibit the flux tracking pattern. The late time broad-band photometric data set can be explained within the framework of the external forward shock model with νm < νc < νx (where νm, νc, and νx are the synchrotron peak, cooling-break, and X-ray frequencies, respectively) spectral regime supporting a rarely observed hard electron energy index (p < 2). We find moderate values of host extinction of E(B − V) = 0.14 ± 0.01 mag for the small magellanic cloud extinction law. In addition, we also report late-time optical observations with the 10.4 m Gran Telescopio de Canarias placing deep upper limits for the host galaxy (z = 1.937), favouring a faint, dwarf host for the burst.