On the solution of the errors in variables problem using the \(l_ 1\) norm (Q1182622)

Let observations \((x_ i,y_ i)\), \(i=1,\dots,m\) and an appropriate model \(y=f(x,a)\) for the data be given where \(a\in \mathbb{R}^ n\), \(n<m\), is a vector of parameters. It is supposed that errors occur in all variables: \(y_ i+r_ i=f(x_ i+\varepsilon_ i,a)\), \(i=1,\dots,m\). The approximation is based on an \(\ell_ 1\)-norm criterion, the components of \(a\) are chosen to minimalize \(\| r\|_ 1+\| \varepsilon \|_ 1\).