# 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

 (25)

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

Proof. It readily follows since and hence

Because and are fixed, is minimized if is maximal over all .

The original OBMP selection criterion proposed in [11] selects the index as the maximizer over of

This condition was proposed in [11] based on the consistency principle which states that the reconstruction of a signal should be self consistent in the sense that, if the approximation is measured with the same vectors, the same measures should be obtained (see [2,3]). Accordingly, the above OBMP criterion was derived in [11] in order to select the measurement vector producing the maximum 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

 (26)

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

Property 3   The index satisfying (25) maximizes over the relative consistency error (26)

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

Hence, maximization of over is equivalent to (25).

It is clear at this point that the forward selection of indices prescribed by proposition (25) is equivalent to selecting the indices by applying OOMP [12] on the projected signal using the dictionary . The routine for implementing the pursuit strategy for subspace selection according to criterion (25) is OBMP.m. An example of application is given in .