# Highly Nonlinear Approximations for Sparse Signal Representation

## Implementing corrections

Let us discuss now the possibility of correcting bad moves in the forward selection, which is specially necessary when dealing with ill posed problems. Indeed, assume we are trying to approximate a signal which is -sparse in a given dictionary, and the search for the right atoms become ill posed after the iteration, , say, with . The -value just indicates that it is not possible to continue with the forward selection, because the computations would become inaccurate and unstable. Hence, if the right solution was not yet found, one needs to implement a strategy accounting for the fact that it is not feasible to compute more than measurement vectors. A possibility is provided by the swapping-based refinement to the OOMP approach introduced in [13]. It consists of interchanging already selected atoms with nonselected ones.Consider that at iteration the correct subspace has not appeared yet and the selected indices are labeled by the indices . In order to choose the index of the atom that minimizes the norm of the residual error as passing from approximation to approximation we should fix the index of the atom to be deleted, say, as the one for which the quantity

is minimized [13,14].

The process of eliminating one atom from the atomic decomposition
is called *backward step* while the process of
adding one atom is called *forward step*.
The forward selection criterion to choose the atom to replace the
one eliminated in the previous step is accomplished by
finding the index
for which
the functional

is maximized. In our framework, using (22), the projector is computed as