On the Significance of Nongeneric Total Least Squares Problems

Abstract

Total least squares (TLS) is one method of solving overdetermined sets of linear equations $AX \approx B$ that is appropriate when there are errors in both the observation matrix B and the data matrix A. Golub was the first to introduce this method into the field of numerical analysis and to develop an algorithm based on the singular value decomposition. However, as pointed out by Golub and Van Loan, some TLS problems fail to have a solution altogether. Van Huffel and Vandewalle described the properties of these so-called nongeneric problems and proposed an extension of the generic TLS problem, called nongeneric TLS, in order to make these problems solvable. They proved that the solution of these nongeneric TLS problems is still optimal with respect to the TLS criteria for any number of observation vectors in B if additional constraints are imposed on the TLS solution space. These constraints are further scrutinized in this paper and compared with other approaches in linear regression.