A new mixed integer programming approach for inverse correspondence analysis

Correspondence analysis (CA) is a dimension reduction technique for categorical data in a two-way contingency table. The low-dimensional CA solution optimally depicts the relationship between the categorical variables. We consider the inverse correspondence analysis (ICA) problem, where we use the low-dimensional representation in order to retrieve the original data table. We propose a mixed integer programming formulation for the ICA problem based on transition formulas, which link the row and column coordinates in a CA solution. We show that our formulation has better theoretical characteristics than the existing formulation in the literature and is able to model a generalised ICA problem, which requires less input. In addition, we introduce an iterative method, which uses the fit of individual points in the low-dimensional CA solution. By incorporating statistical information on the quality of CA solutions into our methodology, we are able to retrieve the original data for larger ICA instances.