Acceleration of Convergence of the Alternating Least Squares Algorithm for Nonlinear Principal Component Analysis

Abstract

Nonlinear principal component analysis (PCA) requires iterative computation using the alternating least squares (ALS) algorithm, which alternates between optimal scaling for quantifying qualitative data and the analysis of optimally scaled data using the ordinary PCA approach. PRINCIPALS of Young et al. (Psychometrika 43:279–281) (1978) and PRINCALS of Gifi (Nonlinear Multivariate Analysis. Wiley, Chichester) (1990) are the ALS algorithms used for nonlinear PCA. When applying nonlinear PCA to very large data sets of numerous nominal and ordinal variables, the ALS algorithm may require many iterations and significant computation time to converge. One reason for the slow convergence of the ALS algorithm is that the speed of convergence is linear. In order to accelerate the convergence of the ALS algorithm, Kuroda et al. (Comput Stat Data Anal 55:143–153) (2011) developed a new iterative algorithm using the vector $\varepsilon$ algorithm by Wynn (Math Comput 16:301–322) (1962).

The original version of this chapter was revised: Typos were corrected throughout the chapter. The erratum to this chapter is available at http://dx.doi.org/10.1007/978-981-10-0159-8_8.