Beyond Euclidean Space: Optimizing Hierarchical Data in Hyperbolic HSVMs

Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Related Works Convex Relaxation Techniques for Hyperbolic SVMs 3.1 Preliminaries 3.2 Original Formulation of the HSVM 3.3 Semidefinite Formulation 3.4 Moment-Sum-of-Squares Relaxation Experiments 4.1 Synthetic Dataset 4.2 Real Dataset Discussions, Acknowledgements, and References Related Works Related Works Related Works Convex Relaxation Techniques for Hyperbolic SVMs 3.1 Preliminaries 3.2 Original Formulation of the HSVM 3.3 Semidefinite Formulation 3.4 Moment-Sum-of-Squares Relaxation Convex Relaxation Techniques for Hyperbolic SVMs 3.1 Preliminaries 3.1 Preliminaries 3.2 Original Formulation of the HSVM 3.2 Original Formulation of the HSVM 3.3 Semidefinite Formulation 3.3 Semidefinite Formulation 3.4 Moment-Sum-of-Squares Relaxation 3.4 Moment-Sum-of-Squares Relaxation Experiments 4.1 Synthetic Dataset 4.2 Real Dataset Experiments Experiments 4.1 Synthetic Dataset 4.1 Synthetic Dataset 4.2 Real Dataset 4.2 Real Dataset Discussions, Acknowledgements, and References Discussions, Acknowledgements, and References Discussions, Acknowledgements, and References A. Proofs A. Proofs B. Solution Extraction in Relaxed Formulation B. Solution Extraction in Relaxed Formulation C. On Moment Sum-of-Squares Relaxation Hierarchy C. On Moment Sum-of-Squares Relaxation Hierarchy D. Platt Scaling [31] D. Platt Scaling [31] E. Detailed Experimental Results E. Detailed Experimental Results F. Robust Hyperbolic Support Vector Machine F. Robust Hyperbolic Support Vector Machine 3.2 Original Formulation of the HSVM Cho et al. [4] proposed the hyperbolic support vector machine which finds a max-margin separator where margin is defined as the hyperbolic point to line distance. We demonstrate our results in a binary classification setting. Extension to multi-class classification is straightforward using Platt-scaling [31] in the one-vs-rest scheme or majority voting in one-vs-one setting. Authors: (1) Sheng Yang, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA (shengyang@g.harvard.edu); (2) Peihan Liu, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA (peihanliu@fas.harvard.edu); (3) Cengiz Pehlevan, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, Center for Brain Science, Harvard University, Cambridge, MA, and Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, MA (cpehlevan@seas.harvard.edu). Authors: Authors: (1) Sheng Yang, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA (shengyang@g.harvard.edu); (2) Peihan Liu, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA (peihanliu@fas.harvard.edu); (3) Cengiz Pehlevan, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, Center for Brain Science, Harvard University, Cambridge, MA, and Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, MA (cpehlevan@seas.harvard.edu). This paper is available on arxiv under CC by-SA 4.0 Deed (Attribution-Sharealike 4.0 International) license. This paper is available on arxiv under CC by-SA 4.0 Deed (Attribution-Sharealike 4.0 International) license. available on arxiv available on arxiv