# 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
- iv)
- Same as in iii) but .

(10) |

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.*

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