Establishment of Information Physics based on algebraic geometry

几乎所有用于信息科学的学习机都是奇异的，因为它们具有退化的Fisher信息矩阵。正则统计理论不能应用于奇异学习机。在这项研究中，我们建立了基于高维代数几何的奇异学习机的新的信息物理理论，并阐明了推广和训练误差的渐近行为是由两个双不变量决定的。基于新的理论，我们导出了一个普遍适用的信息准则，该准则可以从奇异和规则学习机的训练误差中估计泛化误差。

Almost all learning machines used in information science are singular because they have degenerate Fisher Information matrices. Regular statistical theory can not be applied to singular learning machines. In this research, we established new information physics theory for singular learning machines based on higher dimensional algebraic geometry, and clarified that the asymptotic behaviors of generalization and training errors are determined by two birational invariants. Based on new theory, we derived a widely applicable information criterion, by which the generalization error can be estimated from training error in both singular and regular learning machines.