Sheng Gao, Zongming Ma.
Year: 2023, Volume: 24, Issue: 135, Pages: 1−61
Generalized correlation analysis (GCA) is concerned with uncovering linear relationships across multiple data sets. It generalizes canonical correlation analysis that is designed for two data sets. We study sparse GCA when there are potentially multiple leading generalized correlation tuples in data that are of interest and the loading matrix has a small number of nonzero rows. It includes sparse CCA and sparse PCA of correlation matrices as special cases. We first formulate sparse GCA as a generalized eigenvalue problem at both population and sample levels via a careful choice of normalization constraints. Based on a Lagrangian form of the sample optimization problem, we propose a thresholded gradient descent algorithm for estimating GCA loading vectors and matrices in high dimensions. We derive tight estimation error bounds for estimators generated by the algorithm with proper initialization. We also demonstrate the prowess of the algorithm on a number of synthetic data sets.