| 研究実績の概要 |
The sparse modification of the generalized eigenvectors has attracted attention as an effective technique to promote interpretability of many analyses, based on given correlation matrices, e.g., the principal component analysis, the canonical correlation analysis and the Fisher's discriminant analysis, in modern data sciences. However, in reality, such correlation matrices are often estimated in online manner, and therefore the estimation of the generalized eigenvectors as well as their sparse modification must be achieved simultaneously. In this study of 2018, to establish such an adaptive algorithm, we propose an l1-penalized adaptive normalized quasi-Newton algorithm combined with a certain dimension reduction technique. We also present a convergence analysis of the proposed algorithm. Numerical experiments demonstrate that (i) the proposed algorithm with a fixed l1 penalty shows superior steady state performance compared with a straightforward adaptive implementation of an existing algorithm specialized for the sparse PCA, and (ii) the decaying l1 penalty accelerates convergence to the generalized eigenvectors of which sparsity can be assumed, e.g., in the scenario of gene expression analysis. We also studied, as a closely related problem, signal recovery of certain piecewise continuous signals from noisy observations. Fortunately, by establishing newly a tight-dimensional linear transformation which reveals a certain hidden sparsity in discrete samples of finite-dimensional piecewise continuous (FPC) signals, we developed a powerful scheme for the recovery problem.
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