Missing Data, Centering, and Constraints

Abstract

Linear model identification from data with missing values is posed in Sect. 5.1 as a weighted low rank approximation problem with weights related to the missing values equal to zero. Alternating projections and variable projections methods for solving the resulting problem are outlined and implemented. The methods are evaluated on synthetic data and real data from the MovieLens data sets.Low rank approximation is a data modeling tool. Data centering, on the other hand, is a common preprocessing step. The combination of low rank approximation with data centering is studied in Sect. 5.2. In the case of approximation in the Frobenius norm (uniform weighting) and no constraints apart from the rank constraint, the common preprocessing practice of mean subtraction leads to optimal results. Preprocessing by mean subtraction, however, is not the only way to optimally preprocess the data. In the case of approximation by a weighted norm and/or structure constraints on the approximation, preprocessing by mean subtraction leads, in general, to suboptimal results. We show, how classical solution methods for weighted and structured low rank approximation can be modified for doing optimal preprocessing at the same time as low rank approximation.The problem of solving approximately in the least squares sense an overdetermined system of linear equations with complex valued coefficients, where the elements of the solution vector are constrained to have the same phase is reduced in Sect. 5.3 to a generalized low rank matrix approximation.An approximate rank revealing factorization problem with structure constraints on the normalized factors is considered. Examples of structure, motivated by an application of the problem in microarray data analysis, are sparsity, nonnegativity, periodicity, and smoothness. An alternating projections algorithm is developed. Although the algorithm is motivated by a specific application in microarray data analysis, the approach is applicable to other types of structure.