# Highly Nonlinear Approximations for Sparse Signal Representation

## Oblique Matching Pursuit (OBMP)

The criterion we use for the forward recursive selection of the set yielding the right signal separation is in line with the*consistency principle*introduced in [2] and extended in [3]. Furthermore, it happens to coincide with the Optimize Orthogonal Matching Pursuit (OOMP) [12] approach applied to find the sparse representation of the projected signal using the dictionary

By fixing , at iteration we select the index such that is minimized.

**Proposition 8**

*Let us denote by the set of indices Given , the index corresponding to the atom for which is minimal is to be determined as*

*with and the set of indices that have been previously chosen to determine .*

*Proof*. It readily follows since and hence

*consistency error*, with regard to a new measurement . However, since the measurement vectors are not normalized to unity, it is sensible to consider the consistency error relative to the corresponding vector norm , and select the index so as to maximize over the

*relative consistency error*

In order to cancel this error, the new approximation is constructed accounting for the concomitant measurement vector.

*Proof*. Since for all vector given in (19) and we have