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

点击翻译按钮获取中文摘要

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.