text: ---
  author:
  - |
      Ines Chami$^1$[^1][$\ $,]{} Adva Wolf$^1$[,]{} Da-Cheng Juan$^2$[,]{} Frederic Sala$^1$[,]{} Sujith Ravi$^3$[^2]$\ \ $[and]{} Christopher Ré[$^1$]{}\
      $^1$Stanford University\
      $^2$Google Research\
      $^3$Amazon Alexa\
      `{chami,advaw,fredsala,chrismre}@cs.stanford.edu`\
      `dacheng@google.com`\
      `sravi@sravi.org`
  bibliography:
  - 'references.bib'
  title: ' Low-Dimensional Hyperbolic Knowledge Graph Embeddings'
  ---
  
  Introduction
  ============
  
  Related Work
  ============
  
  Problem Formulation and Background
  ==================================
  
  We describe the KG embedding problem setting and give some necessary background on hyperbolic geometry.
  
  Knowledge graph embeddings
  --------------------------
  
  Hyperbolic geometry {#subsec:hyp_geom}
  -------------------
  
  Methodology
  ===========
  
  Hierarchies in hyperbolic space {#subsec:hypmodel}
  -------------------------------
  
  Hyperbolic isometries {#subsec:isom}
  ---------------------
  
  Hyperbolic attention {#subsec:attention}
  --------------------
  
  The [<span style="font-variant:small-caps;">AttH</span>]{} model {#subsec:fullmodel}
  ----------------------------------------------------------------
  
  Experiments {#subsec:LP}
  ===========
  
  Experimental setup
  ------------------
  
  Results in low dimensions {#sec:explowdim}
  -------------------------
  
  Hyperbolic rotations and reflections {#sec:expops}
  ------------------------------------
  
  Attention-based transformations {#sec:expatt}
  -------------------------------
  
  Results in high dimensions {#subsec:high_dim}
  --------------------------
  
  Visualizations
  --------------
  
  Conclusion
  ==========
  
  Acknowledgements {#acknowledgements .unnumbered}
  ================
  
  Appendix
  ========
  
  Below, we provide additional details. We start by providing the formula for the hyperbolic analogue of addition that we use, along with additional hyperbolic geometry background. Next, we provide more information about the metrics that are used to determine how hierarchical a dataset is. Afterwards, we give additional experimental details, including the table of hyperparameters and further details on tangent space optimization. Lastly, we include an additional comparison against the Dihedral model [@xu2019relation].
  
  Möbius addition {#sec:appendix_hyp}
  ---------------
  
  Hierarchy estimates {#subsec:curvature_est}
  -------------------
  
  Experimental details {#subsec:appendix_param}
  --------------------
  
  Tangent space optimization {#subsec:tangent_optim}
  --------------------------
  
  Comparison to Dihedral
  ----------------------
  
  [^1]:   Work partially done during an internship at Google.
  
  [^2]:   Work done while at Google AI.
  