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

核克矩阵的Nystrom近似法分两步降低了每个矩阵的秩。为了有效地近似这些矩阵，重要的是为每一步设置适当的约简率，并处理近似精度和计算成本之间的权衡。在本研究项目中，我们采用计算物理的方法对大规模随机矩阵进行分析，并优化了约简率。实验结果表明，该方法在较低的计算成本下，对真实的手写体数据具有较高的识别精度。给出了Nystrom方法近似误差的上界，并证明了其统计一致性。该方法不仅适用于Nystrom方法，还适用于稀疏贪婪近似和不完全Cholesky分解等近似方法。与此同时，我们研究了交换代数统计，并将计算代数应用于渐近估计理论，提出了一种新的统计估计量。此外，我们提出了一种统计方法来分析心理词典的树形图，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.