Sensitivity and conditioning of the truncated total least squares solution (Q2866235)

The paper presents an explicit expression for the condition number of the truncated total least square (T-TLS) solution of NEWLINE\[NEWLINE Ax \approx b, NEWLINE\]NEWLINE where \(A\in \mathbb R^{m \times n}\), \(b\in \mathbb R^m\) are not precisely known. The T-TLS solution with truncation level \(k\) is the minimum \(2\)-norm solution of \(A_k x = b_k\), where \([A_k,b_k]\) is the best rank-\(k\) approximation of \([A,b]\) in the Frobenius norm. This method uses the singular value decomposition of the augmented matrix: NEWLINE\[NEWLINE [A,b]=U\Sigma V^T. NEWLINE\]NEWLINE The singular values of \([A,b]\) are diagonal entries of \(\Sigma \in \mathbb R^{m \times (n+1)}\), denoted by \(\sigma_i\), sorted in nonincreasing order. The truncation level \(k<\min\{m,n+1\}\) is such that \(\sigma_k>\sigma_{k+1}\). The T-TLS solution is given by an explicit formula using an appropriate partition of \(V\) based on \(k\).NEWLINENEWLINEThe condition number of the T-TLS problem is obtained by the Fréchet derivative. Upper bounds on this condition number are derived. They are simple to compute and to interpret. These results generalize those in the literature for untruncated TLS problems. Numerical experiments demonstrate that the bounds are a very good estimate of the condition number and provide a significant improvement to known bounds.NEWLINENEWLINEThe paper is well-written and of high presentation quality. It is based on the matrix analysis exploring the matrix tensor product, the vec-operation and other sophisticated linear algebra techniques. On the other hand, it is simple to understand. The numerical experiments are impressive and show that the bounds are tinny. The paper can be recommended to all, how are interested in the sensitivity of the T-TLS solution.