# Data Augmentation for Mathematical Objects

arXiv: https://arxiv.org/abs/2307.06984 · HTML: https://mlsanity.com/paper/2307.06984/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 13 Jul 2023.

## Builds on (most cited first, 9 of 9)

- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/1404.6369/) (2014)
- [Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/1904.11061/) (2019)
- [Algorithmically generating new algebraic features of polynomial systems for machine learning](https://mlsanity.com/paper/1906.01455/) (2019)
- [Improved cross-validation for classifiers that make algorithmic choices to minimise runtime without compromising output correctness](https://mlsanity.com/paper/1911.12672/) (2019)
- [A machine learning based software pipeline to pick the variable ordering for algorithms with polynomial inputs](https://mlsanity.com/paper/2005.11251/) (2020)
- [TheoryGuru: A Mathematica Package to apply Quantifier Elimination](https://mlsanity.com/paper/1806.10925/) (2018)
- [Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driven Search Using Cylindrical Algebraic Coverings](https://mlsanity.com/paper/2003.05633/) (2020)
- [Revisiting Variable Ordering for Real Quantifier Elimination using Machine Learning](https://mlsanity.com/paper/2302.14038/) (2023)

## Built on by (most cited first, 2 of 2)

- [Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD](https://mlsanity.com/paper/2401.13343/) (2024)
- [Breaking the Data Barrier in Learning Symbolic Computation: A Case Study on Variable Ordering Suggestion for Cylindrical Algebraic Decomposition](https://mlsanity.com/paper/2601.13731/) (2026)

## Abstract (arXiv)

This paper discusses and evaluates ideas of data balancing and data augmentation in the context of mathematical objects: an important topic for both the symbolic computation and satisfiability checking communities, when they are making use of machine learning techniques to optimise their tools. We consider a dataset of non-linear polynomial problems and the problem of selecting a variable ordering for cylindrical algebraic decomposition to tackle these with. By swapping the variable names in already labelled problems, we generate new problem instances that do not require any further labelling when viewing the selection as a classification problem. We find this augmentation increases the accuracy of ML models by 63% on average. We study what part of this improvement is due to the balancing of the dataset and what is achieved thanks to further increasing the size of the dataset, concluding that both have a very significant effect. We finish the paper by reflecting on how this idea could be applied in other uses of machine learning in mathematics.