An Analysis of the Total Approximation Problem in Separable Norms, and an Algorithm for the Total $l_1 $ Problem

In many data fitting problems there are errors in the $m \times n$ data matrix M as well as in the observed vector ${\bf f} \in R^m $. It is possible to take account of this by formulating and solving the total approximation problem in which some norm of the $m \times (n + 1)$ error matrix is minimized. For a general class of matrix norms, which we call separable, it is shown that the solution is a rank one matrix, and that the problem may be solved when the solution is known to a vector norm minimization problem on $R^m $ with a single equality constraint. Attention is focussed on the total $l_1 $ problem. A finite descent algorithm is developed, and numerical results illustrating it are given.