Highly Nonlinear Approximations for Sparse Signal Representation

Possible constructions of oblique projector

Notice that the oblique projector onto is independent of the selection of the spanning set for . The possibility of using different spanning sets yields a number of different ways of computing , all of them theoretically equivalent, but not necessarily numerically equivalent when the problem is ill posed.

Given the sets and we have considered the following theoretically equivalent ways of computing vectors .

i) , where is the -th element of the inverse of the matrix having elements .
ii)
Vectors are as in i) but the matrix elements of are computed as .
iii)
Orthonormalising to obtain vectors are then computed as with .
iv)
Same as in iii) but .
Moreover, considering that , are the eigenvectors of and assuming that there exist nonzero eigenvalues on ordering these eigenvalues in descending order , we can express the matrix elements of the Moore-Penrose pseudo inverse of as: (10)

with the -th component of . Moreover, the orthonormal vectors (11)

are singular vectors of , which satisfies , as it is immediate to verify. By defining now the vectors as (12)

the projector in (3) is recast (13)

Inversely, the representation (3) of arises from (13), since (14)

Proposition 2   The vectors and given in (11) and (12) are biorthogonal to each other and span and , respectively.

The proof of this proposition can be found in  Appendix A.

All the different numerical computations for an oblique projector discussed above can be realized with the routine ObliProj.m.