Sparse factor analysis for categorical data with the group-sparse generalized singular value decomposition

Correspondence analysis, multiple correspondence analysis, and their discriminant counterparts (i.e., discriminant simple correspondence analysis and discriminant multiple correspondence analysis) are methods of choice for analyzing multivariate categorical data. In these methods, variables are integrated into optimal components computed as linear combinations whose weights are obtained from a generalized singular value decomposition (GSVD) that integrates specific metric constraints on the rows and columns of the original data matrix. The weights of the linear combinations are, in turn, used to interpret the components, and this interpretation is facilitated when components are 1) pairwise orthogonal and 2) when the values of the weights are either large or small but not intermediate—a configuration called a simple or a sparse structure. To obtain such simple configurations, the optimization problem solved by the GSVD is extended to include new constraints that implement component orthogonality and sparse weights. Because multiple correspondence analysis represents qualitative variables by a set of binary columns in the data matrix, an additional group constraint is added to the optimization problem in order to sparsify the whole set of columns representing one qualitative variable. This method—called group-sparse GSVD (gsGSVD)—integrates these constraints in a new algorithm via an iterative projection scheme onto the intersection of subspaces where each subspace implements a specific constraint. This algorithm is described in details, and we show how it can be adapted to the sparsification of simple and multiple correspondence analysis (as well as their barycentric discriminant analysis versions). This algorithm is illustrated with the analysis of four different data sets—each illustrating the sparsification of a particular CA-based method.