# Machine learning the real discriminant locus

arXiv: https://arxiv.org/abs/2006.14078 · HTML: https://mlsanity.com/paper/2006.14078/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2020.

## Built on by (most cited first, 5 of 5)

- [On the Instability of Relative Pose Estimation and RANSAC's Role](https://mlsanity.com/paper/2112.14651/) (2021)
- [Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/2304.12154/) (2023)
- [Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD](https://mlsanity.com/paper/2401.13343/) (2024)
- [Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC](https://mlsanity.com/paper/2310.02719/) (2023)
- [Constrained Neural Networks for Interpretable Heuristic Creation to Optimise Computer Algebra Systems](https://mlsanity.com/paper/2404.17508/) (2024)

## Abstract (arXiv)

Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. For multidimensional parameter spaces, this article presents a novel sampling method which carefully samples the parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbor and deep learning are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks the real solution paths unlike traditional methods which track all~complex~solution~paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the Kuramoto model.