图拉普拉斯算子、黎曼流形及其机器学习
Graph Laplacians, Riemannian Manifolds and their Machine-Learning
摘要：
图拉普拉斯算子以及相关的谱不等式和（上）同调为黎曼流形的离散类似物提供了一个尝试，提供了组合学，几何学和理论物理学之间的丰富的相互作用。我们将数据科学中的一些最新技术，如监督和无监督机器学习和拓扑数据分析应用于Wolfram数据库中的8000个有限图，以研究这些对应关系。令人鼓舞的是，我们发现，神经分类器，回归器和网络可以执行，高效率和准确性，众多的任务，从识别图的Ricci平坦，预测谱隙，检测哈密顿圈的存在，等等。
英文摘要：
Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics. We apply some of the latest techniques in data science such as supervised and unsupervised machine-learning and topological data analysis to the Wolfram database of some 8000 finite graphs in light of studying these correspondences. Encouragingly, we find that neural classifiers, regressors and networks can perform, with high efficiently and accuracy, a multitude of tasks ranging from recognizing graph Ricci-flatness, to predicting the spectral gap, to detecting the presence of Hamiltonian cycles, etc.
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