Geometric Generality of Transformer-Based Gröbner Basis Computation
This work provides a rigorous geometric foundation for applying Transformers to symbolic mathematics, addressing a key challenge in dataset generation for this domain.
The paper tackles the problem of ensuring the generality and quality of datasets for training Transformers to compute Gröbner bases, proving that a previously proposed dataset generation method is sufficiently general and extending it to enhance training effectiveness.
The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gröbner basis using Transformers. While a dataset generation method tailored to Transformer-based Gröbner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gröbner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.