Condition numbers for the truncated total least squares problem and their estimations

Abstract

In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of Ax ≈ b under the genericity condition, where A is a m × n real data matrix and b is a real m-vector. Moreover, we reveal that normwise, componentwise, and mixed condition numbers for the TTLS problem can recover the previous corresponding counterparts for the total least squares (TLS) problem when the truncated level of the TTLS problem is n. When A is a structured matrix, the structured perturbations for the structured truncated TLS (STTLS) problem are investigated and the corresponding explicit expressions for the structured normwise, componentwise, and mixed condition numbers for the STTLS problem are obtained. Furthermore, the relationships between the structured and unstructured normwise, componentwise, and mixed condition numbers for the STTLS problem are studied. We devise reliable condition estimation algorithms for the TTLS problem by utilizing small-sample statistical condition estimation techniques. The proposed condition estimation algorithms employ the singular value decomposition (SVD) of the augmented matrix [A b] to reduce the computational complexity, where both unstructured and structured normwise, mixed, and componentwise condition estimations are considered. The proposed condition estimation algorithms can be integrated into the SVD-based direct solver for the small and medium size TTLS problem to give the error estimation for the numerical TTLS solution. Numerical experiments are reported to illustrate the reliability of the proposed condition estimation algorithms.

Structured condition numbers for the total least squares problem with linear equality constraint and their statistical estimation

“…We use two algorithms to estimate the unstructured normwise, mixed and componentwise condition number. The first one, outlined in Algorithm 1, is based on the SSCE (small-sample statistical condition estimation) method [29] and has been applied to estimate the unstructured normwise condition number (see [31,32,33,16,17]). The second one, outlined in Algorithm 2, is also from [29] and has been used to estimate the unstructured mixed and componentwise condition numbers.…”

“…To estimate the structured normwise, mixed and componentwise condition numbers, we use Algorithm 3 which is also a SSCE method. The algorithm is from [29] and has been applied to many problems (see e.g., [30,16,17]).…”

“…For structured TLS problems, it is suitable to investigate structured perturbations on the input data, because structure-preserving algorithms that preserve the underlying matrix structure can enhance the accuracy and efficiency of the TLS solution computation. Recently, structured condition numbers of TLS problem, the mixed LS-TLS problem, and truncated-TLS problem have been considered; see [14,15,16,17,18] However, to our best knowledge, there is no work on structured perturbation analysis and structured condition numbers of TLSE problem so far. Therefore, in this paper, we study structured condition numbers of TLSE problem, including the normwise, mixed, and componentwise condition number and their relationships corresponding to the unstructured ones.…”

Quantum-inspired algorithm for truncated total least squares solution

“…The more relationships between the LS and TLS problems can be seen in [1,2,7]. Besides, for the condition numbers of the total least squares problem have been considered in [8][9][10][11][12][13][14][15][16]. The concept of the core problem is proposed by Paige and Strakoš in [17] and used to find the minimum norm solution of the TLS problem.…”

Strong consistency of the projected total least squares dynamic mode decomposition for datasets with random noise

“…A typical example is a rank-constrained TLS problem where the coefficient matrix A has collinearity. This problem can be solved by the so-called truncated SVD [6,12,21], and statistical consistency is presented in [24]. A similar constraint, the reduced rank estimation, which assumes low rank of X, is also important and often used [22].…”