Highly Nonlinear Approximations for Sparse Signal Representation
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 .
- , where is the -th element of the inverse of the matrix having elements .
- Vectors are as in i) but the matrix elements of are computed as .
then computed as
- Same as in iii) but .
with the -th component of . Moreover, the orthonormal vectors
are singular vectors of , which satisfies , as it is immediate to verify. By defining now the vectors as
the projector in (3) is recast
Inversely, the representation (3) of arises from (13), since
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.