Platt Scaling for HSVM: Calibrating Binary to Probabilistic Predictions

Table of Links

Abstract and 1. Introduction

2. Related Works

3. 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

4. Experiments

   4.1 Synthetic Dataset

   4.2 Real Dataset

5. Discussions, Acknowledgements, and References

A. Proofs

B. Solution Extraction in Relaxed Formulation

C. On Moment Sum-of-Squares Relaxation Hierarchy

D. Platt Scaling [31]

E. Detailed Experimental Results

F. Robust Hyperbolic Support Vector Machine

D Platt Scaling [31]

Platt scaling [31] is a common way to calibrate binary predictions to probabilistic predictions in order to generalize binary classification to multiclass classification, which has been widely used along with SVM. The key idea is that once a separator has been trained, an additional logistic regression is fitted on scores of the predictions, which can be interpreted as the closeness to the decision boundary.

In the context of HSVM, suppose w∗ is the linear separator identified by the solver, then we find two scalars, 𝐴, 𝐵 ∈ R, with

where ∗ refers to the Minkowski product defined in Equation (1). The value of 𝐴 and 𝐵 are trained on the trained set using logistic regression with some additional empirical smoothing. For one-vs-rest training, we will then have 𝐾 sets of (𝐴, 𝐵) to train, and at the end we classify a sample to the class with the highest probability. See detailed implementation here https://home.work.caltech.edu/ htlin/program/libsvm/doc/platt.py in LIBSVM.