Efficiency for machine to learn discrete geometric objects

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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.