Summary
This paper applies machine learning methods to predict whether an algebraic variety is Q-Fano. Q-fano varieties are Fano varieties that are Q-factorial and have terminal singularities. Checking terminally is computational challenging. To train a machine learning classifier, the authors build a balanced dataset of 10 million eight-dimensional Q-factorial Fano toric varieties of Picard rank 2. They use an MLP to classify whether a topic variety has terminal singularities (and thus a Q-fano variety) from its weight matrix. The neural network achieves an accuracy of 95%, which leads to the speculation that these exists a simple criterion to check terminal singularities directly from the weight matrix. The authors develops a new algorithm that is 15 times faster than the naive approach but still orders of magnitude slower than the ML classifier. They then conclude that the ML results suggest that a simpler criterion exists. Facilitated by the faster ML model, the authors create and visualize a dataset of 100 million examples, which provides new insights for the patterns in the classification.
Strengths
- This paper provides an excellent example that shows how ML can be used to guide research in mathematics. The authors use neural networks to accelerate the classification of Fano-varieties. The computation speed by neural networks motivates new classification algorithms and suggests room for even faster ones. The trained ML model is then used as a surrogate to generate large amount of data which would be prohibitively expensive using traditional methods, and provide insights that guides theoretical understandings. While ML for science is getting popular in recent years, ML for mathematics is still a new field but has immense potential.
- The Q-factorial Fano dataset provided in this paper is a valuable assets for both math and ML communities. It will advance the understanding of Q-Fano classification and benefit future development on theorem discovery.
- The last section provides a number of interesting future directions. By relating the limitations of the current method to fundamental questions in ML (out-of-distribution generalization, data efficiency, geometric deep learning, interpretability), the authors demonstrate the importance of these topics and paves way for a new area in ML research.
Weaknesses
- The mathematical objects studied in this paper is not very accessible to people without a math background. As a result, the significance of results, such as the analysis of invariant of Fano-varieties section 6, is not as strong as it could be. This is perhaps due to the large amount of prerequisite for algebraic geometry, but the authors could include more introductory materials, for example, in the appendix.
- While the AI-assisted workflow is novel, the contribution on machine learning is limited. In particular, the neural network uses multilayer perceptron, and no justification is given for the architecture choice. The theoretical result in section 5 may not be interesting to a NeurIPS audience.
- The claim that “the ML results suggests that a simpler criterion exists” for Q-Fano classification (line 82) is not well supported. The ML model does not have 100% accuracy and does not generalize when weights matrices fall outside of the range used in training. Hence the provided results does not prove that the ML model has found a simpler and correct criterion.
Questions
- Would it be possible for the authors to comment more on the significance of the classification of Q-Fano varieties?
- Does detecting terminal singularities have other applications beyond a criterion for Q-Fano varieties?
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
The authors point out limitations in Section 7. There are no potential negative societal impacts of the work.