NUMERICAL RIEMANNIAN OPTIMIZATION FOR HIERARCHICAL EMBEDDINGS ON THE POINCARE´ DISK

• 발행기관: 충청수학회
• 저자명: Jong Ryul Kim
• 간행물 정보: 『Journal of the Chungcheong Mathematical Society』Volume 39, No. 2, 97~108쪽, 전체 12쪽
• 주제분류: 자연과학 > 자연과학일반
• 파일형태: PDF
• 발행일자: 2026.05.30

국문 초록

This study explores the mathematical foundations and applications of the Poincar´e disk model for hierarchical data rep- resentation. Unlike Euclidean embeddings, which often suffer from structural distortion due to polynomial space growth, hyperbolic ge- ometry naturally accommodates the exponential expansion of hier- archical nodes. We derive the M¨obius addition from complex linear fractional transformations and extend it to a vector-based frame- work within Ungar’s gyrovector space. By rigorously formulating the Riemannian metric and geodesics, we implement a Riemannian gradient descent algorithm using M¨obius-based retractions. Our re- sults, optimized via Poincar´e Negative Sampling Loss, demonstrate that this geometric approach effectively preserves hierarchical struc- tures by balancing attraction and peripheral repulsion.

목차

1. Introduction

2. Möbius Transformations and Gyrovector Spaces

3. Geodesics and Riemannian Optimization

4. Learning Hierarchical Representations

5. Experimental Results

6. Conclusion

7. Appendix

References

키워드

Poincaré disk Hyperbolic embedding Möbius addi- tion Riemannian SGD Hierarchical representation Numerical geodesics WordNet taxonomy

참고문헌

• University of Minnesota, School of Mathematics
• Proceedings of the 35th International Conference on Machine Learning(ICML)
• MSRI Publications
• Honam Mathematical J.
• J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math.
• J. Chungcheong Math. Soc.
• Advances in Neural Information Processing Systems(NeurIPS)
• World Scientific Publishing Co.