Level choice in truncated total least squares

The method of truncated total least squares [2] is an alternative to the classical truncated singular value decomposition used for the regularization of ill-conditioned linear systems Ax ≈ b [3]. Truncation methods aim at limiting the contribution of noise or rounding errors by cutting off a certain number of terms in an expansion such as the singular value decomposition. To this end a truncation level k must be carefully chosen. The truncated total least squares solution becomes more significantly dominated by noise or errors when the truncation level k is overestimated than the truncated singular value decomposition solution does. Model selection methods that are often applied in the context of the truncated singular value decomposition can be modified to be applied in the context of the truncated total least squares. The proposed modified generalized cross validation combined with the truncated total least squares method performs better than the classical generalized cross validation [1] combined with the truncated singular value decomposition, especially when both the coefficient matrix A and the right-hand side vector b are contaminated by noise.