A Multi-Step Nonlinear Dimension-Reduction Approach with Applications to Big Data

Abstract

In this paper, a novel dimension-reduction approach is presented to overcome challenges such as nonlinear relationships, heterogeneity, and noisy dimensions. Initially, the <inline-formula><tex-math notation= LaTeX >$p$</tex-math><alternatives><mml:math><mml:mi>p</mml:mi></mml:math><inline-graphic xlink:href= raghavan-ieq1-2876848.gif /></alternatives></inline-formula> attributes in the data are first organized into random groups. Next, to systematically remove redundant and noisy dimensions from the data, each group is independently mapped into a low dimensional space via a parametric mapping. The group-wise transformation parameters are estimated using a low-rank approximation of distance covariance. The transformed attributes are reorganized into groups based on the magnitude of their respective eigenvalues. The group-wise organization and reduction process is performed until a user-defined criterion on eigenvalues is satisfied. In addition, novel procedures are introduced to aggregate the transformation parameters when the data is available in batches. Overall performance is demonstrated with extensive simulation analysis on classification by employing 10 data-sets.