Sparse Correspondence Analysis for Contingency Tables

Abstract

Since the introduction of the lasso in regression, various sparse methods have been developed in an unsupervised context like sparse principal component analysis (s-PCA), sparse canonical correlation analysis (s-CCA) and sparse singular value decomposition (s-SVD). These sparse methods combine feature selection and dimension reduction. One advantage of s-PCA is to simplify the interpretation of the (pseudo) principal components since each one is expressed as a linear combination of a small number of variables. The disadvantages lie on the one hand in the difficulty of choosing the number of non-zero coefficients in the absence of a well established criterion and on the other hand in the loss of orthogonality for the components and/or the loadings. In this paper we propose sparse variants of correspondence analysis (CA)for large contingency tables like documents-terms matrices used in text mining, together with pPMD, a deation technique derived from projected deflation in s-PCA. We use the fact that CA is a double weighted PCA (for rows and columns) or a weighted SVD, as well as a canonical correlation analysis of indicator variables. Applying s-CCA or s-SVD allows to sparsify both rows and columns weights. The user may tune the level of sparsity of rows and columns and optimize it according to some criterium, and even decide that no sparsity is needed for rows (or columns) by relaxing one sparsity constraint. The latter is equivalent to apply s-PCA to matrices of row (or column) profiles.