Generalized Eigenvalue Problems: Algorithms and Applications

Nowadays we encounter massive data from various fields, and there are growing demands for efficient methods to transform data into the useful information. Especially, most data from various research areas contain observations of more than two variables, which we call "multivariate data". Many statistical methods have been developed to extract useful information under the category "multivariate data analysis". As dimensionality of the data gets bigger with the development of technologies, many statisticians are striving to keep pace with this change by proposing updated methods that are appropriate for those data. There are at least two characteristics of the recent data. One is high dimensionality, the data gets bigger and bigger. In many cases the number of variables is much larger than the sample size. The other one is the complexity. Multi-dimensional array data, known as a tensor, is emerging. The imaging data is the representative one. Only a few statistical methods exist to handle these data. In this dissertation, we study some tools to deal with the data with these characteristics.;In Chapter 2, we suggest a unifying approach for many constrained multivariate analysis methods. It is well known that many multivariate analysis methods are based on the generalized eigenvalue problem, which can be formulated as an optimization problem. Building on this relationship, we can solve constrained multivariate analysis problem using existing optimization algorithms. We introduced five classes of optimization algorithms to solve the constrained generalized eigenvalue problem and tested their effectiveness on some sparsity or nonnegativity constrained statistical problems.;In Chapter 3, we propose novel Canonical Correlation Analysis (CCA) methods for two multi-dimensional array data sets. There is little literature about multivariate analysis of tensor data, especially there are no methods to conduct CCA on tensor data without destroying its own structure. Our models are based on a decomposition of the tensor parameter that enables us to handle the problem with manageable number of parameters. Also, by imposing parsimonious structure on the covariance matrix, we achieved better statistical efficiency. Simulation studies are conducted to assess performance of our proposed models. The results show that our methods exhibit excellent performance.