Model selection and machine learning theory via large-scale random matrices

核图矩阵的Nystrom逼近法分两步对每个矩阵进行降阶。为了有效地逼近这些矩阵，重要的是为每一步设置适当的缩减率，并处理好逼近精度和计算成本之间的权衡。在这个研究项目中，我们使用计算物理的方法来分析大规模随机矩阵，并优化了约简率。实验结果表明，该方法对手写字符的真实数据具有很高的准确率，而且计算代价很低。给出了Nystrom方法逼近误差的上界，并证明了其统计相合性。此外，该方法不仅可用于Nystrom方法，还可用于稀疏贪婪逼近和不完全Cholesky分解等其他逼近方法。在这项研究的同时，我们研究了交换代数统计量，并将计算代数应用到渐近估计理论中，提出了一种新的统计估计量。此外，我们还提出了一种统计方法来分析心理词典的树状图，这是一个具有代数结构的模型的例子。

Nystrom approximation method for kernel gram matrices reduces the rank of each matrix in two steps. In order to approximate those matrices efficiently, it is important to set an adequate reduction rate for each step and handle the tradeoff between accuracy of the approximation and cost of the computation. In this research program, we used methods of computational physics for analyzing large-scale random matrices and optimized the reduction rates. We checked experimentally that the proposed method attains high accuracy even with very low computational cost for real data of hand-written characters. We derived an upper bound for approximation error of Nystrom method and proved the statistical consistency. Moreover, the proposed method can be used not only for Nystrom method but also for other approximation methods including the sparse greedy approximation and the incomplete Cholesky decomposition. In parallel with this research, we studied commutative algebraic statistics and proposed a novel statistical estimator by applying the computational algebra to the asymptotic estimation theory. In addition, we proposed a statistical method to analyze dendrograms of mental lexicon, which is an example of models holding an algebraic structure.