Efficiency for machine to learn discrete geometric objects

通过研究流形连通分支个数的估计方法，将主成分分析的VC维作为统计学习进行估计。接下来，我们引入了一种新的序型集合系统来衡量学习的难度。我们证明了：(1)任何好的拟序都可以用具有我们的序型的集合系统来表示；(2)如果一个集合系统有我们的序型，那么它的一个连续的Cantor单调函数映象就是我们的序型。最后，我们将集合系统的序型理论发展为更好的拟序理论。

By studying evaluation methods of the number of connected components of manifolds, we evaluated the VC dimensions of principal component analysis, as statistical learning. Next, we introduce a new order type of set systems to measure the difficulty to learn. We proved (1)any well quasi-ordering can be represented by a set system having our order type ; and (2)if a set system has our order type, then a continuous image of it by a Cantor monotone function does so. Finally, we developed the theory of our order type of set systems to the theory of better quasi-orderings.