用神经网络研究纽结不变量的可学习性，发现辫表示最有效。
On the Learnability of Knot Invariants: Representation, Predictability, and Neural Similarity
- 采用辫表示能更好预测纽结不变量
- 双曲几何和纽结图的不变量易学，而同调类不变量难学
- 提出梯度显著性相似度，揭示模型间学习关联
我们分析了神经网络预测纽结不变量的不同方面。首先，研究不同纽结表示对不变量预测的影响，发现辫表示通常表现最佳。其次，考察哪些不变量易于学习，结果表明由双曲几何和纽结图导出的不变量很容易学习，而由拓扑或同调数据导出的不变量则较难学习；预测Arf不变量在任何表示下均无法学习。第三，我们提出一种基于梯度显著性向量的余弦相似度评分，以及联合误分类评分，用于揭示训练预测相关拓扑不变量的神经网络之间的相似性。
原文摘要 · Abstract (English)
We analyze different aspects of neural network predictions of knot invariants. First, we investigate the impact of different knot representations on the prediction of invariants and find that braid representations work in general the best. Second, we study which knot invariants are easy to learn, with invariants derived from hyperbolic geometry and knot diagrams being very easy to learn, while invariants derived from topological or homological data are harder. Predicting the Arf invariant could not be learned for any representation. Third, we propose a cosine similarity score based on gradient saliency vectors, and a joint misclassification score to uncover similarities in neural networks trained to predict related topological invariants.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。