Dealing with Prior Knowledge

Centering by mean subtraction is a common preprocessing step applied before other data modeling methods are used. Section 8.1 studies different ways of combining linear data modeling by low-rank approximation with data centering. It is shown that in the basic case of approximation in the Frobenius norm and no structure, the two-step procedure of preprocessing the data by mean subtraction and low-rank approximation is optimal. In the case of approximation in either a weighted norm or with a structural constraint, the two-step procedure is suboptimal. We show modifications of the variable projection and alternating projections methods for weighted and structured low-rank approximation with centering. Section 8.2 considers an approximate rank revealing factorization problem with structural constraints on the normalized factors. 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. Section 8.3 considers 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. This problem is reduced to a generalized low-rank matrix approximation. Section 8.4 considers a blind identification problem with prior knowledge about the input in the form of a linear time-invariant autonomous system. A special case of the problem for constant input has an application in metrology for dynamic measurement. The general problem can be viewed alternatively as a gray-box identification problem or an input estimation problem with unknown model dynamics.