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 A Proofs A.1 Deriving Soft-Margin HSVM with polynomial constraints This section describe the key steps to transform from Equation (6) to Equation (7) for an efficient implementation in solver as well as theoretical feasibility to derive semidefinite and moment-sum-of-squares relaxations subsequently. By introducing the slack variable 𝜉𝑖 as the penalty term in Equation (6), we can rewrite Equation (6) into then by rearranging terms and taking sinh on both sides, it follows that where the last equality follows from hyperbolic trig identities. To make it ready for moment-sum-of-squares relaxation, we turn the function into a polynomial constraint by taking the taylor expansion up to some odd orders (we need monotonic decreasing approximation to 𝑔(.), so we need odd orders). If taking up to the first order, we relax the problem into Equation (7) [2]. If taking up to the third order, we relax the original problem to, It’s worth mentioning that we expect the lower bound gets tighter as we increase the order of Taylor expansion. However, once we apply the third order Taylor expansion, the constraint is no longer quadratic, eliminating the possibility of deriving a semidefinite relaxation. Instead, we must rely on moment-sum-of-squares relaxation, potentially requiring a higher order of relaxation, which may be highly time-costly. A.2 Stereographic projection maps a straight line on H2 to an arc on Poincaré ball B2 Suppose w = [𝑤0, 𝑤1, 𝑤2] is a valid hyperbolic decision boundary (i.e. w ∗ w < 0), and suppose a point on the Lorentz straight line, 𝑥 = [𝑥0, 𝑥1, 𝑥2] with w ∗ 𝑥 = 0, is mapped to a point, 𝑣 = [𝑣1, 𝑣2].l in Poincaré space, then we have If 𝑤 > 0, we further have One could show that if 𝑤0 = 0, then it is the "arc" of a infinitely large circle, or just a Euclidean straight line passing through the origin with normal vector (𝑤1, 𝑤2). With this simplification, one could plot the decision boundary on the Poincaré ball easily. 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 [2] note that in [4], the authors use 1 − 𝜉𝑖 instead of 1 − √ 2𝜉𝑖 . We consider our formulation less sensitive to outliers than the former formulation.

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