The Numerical Stability of Hyperbolic Representation Learning
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866913623883907072 |
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| author | Mishne, Gal Wan, Zhengchao Wang, Yusu Yang, Sheng |
| author_facet | Mishne, Gal Wan, Zhengchao Wang, Yusu Yang, Sheng |
| contents | Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincaré ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincaré ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincaré ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00181 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Numerical Stability of Hyperbolic Representation Learning Mishne, Gal Wan, Zhengchao Wang, Yusu Yang, Sheng Machine Learning Numerical Analysis Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincaré ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincaré ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincaré ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM. |
| title | The Numerical Stability of Hyperbolic Representation Learning |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2211.00181 |