Two-stage acceleration for non-linear PCA

摘要

Principal components analysis (PCA) is a descriptive multivariate method for analyzing quantitative data. For PCA of a mixture of quantitative and qualitative data, quantification of qualitative data requires obtaining optimal scaling data and using ordinary PCA. The extended PCA, including such quantification, is called non-linear PCA. Then, the alternating least squares (ALS) algorithm is used as the quantification method. However, the ALS algorithm for non-linear PCA of large data requires many iterations and much computation time due to its linear convergence. We provide a new acceleration method for the ALS algorithm using the vector ε (vε) and Graves-Morris (GM) algorithms. Both acceleration algorithms speed up the convergence of a linearly convergent sequence generated by the ALS algorithm. Acceleration of the ALS algorithm can be performed in two stages: 1) the vε algorithm generates an accelerated sequence of the ALS sequence and 2) the convergence of the vε accelerated sequence is accelerated using the GM algorithm. Thus, we expect that, by accelerating the convergence of the vε accelerated sequence, the GM algorithm improves the overall computational efficiency of the ALS algorithm. Numerical experiments examine the performance of the two-stage acceleration for non-linear PCA.