Highly Nonlinear Approximations for Sparse Signal Representation
Possible constructions of oblique projector
Notice that the oblique projector onto![$ {\cal{V}}$](img39.png)
![$ {\cal{W}}$](img184.png)
![$ \hat{E}_{{\cal{V}}{\cal{W}^\bot}}$](img167.png)
Given the sets
and
we have considered the following theoretically equivalent ways of
computing vectors
.
- i)
-
, where
is the
-th element of the inverse of the matrix
having elements
.
- ii)
- Vectors
are as in i) but the matrix elements of
are computed as
.
- iii)
- Orthonormalising
to obtain
vectors
are then computed as
.
- iv)
- Same as in iii) but
.
![$ \psi_n \in \mathbb{F}^M, n=1,\ldots,M$](img294.png)
![$ \hat{G}$](img241.png)
![$ N$](img295.png)
![$ \lambda_n, n=1,\ldots,N$](img296.png)
![$ G$](img211.png)
![]() |
(10) |
with
![$ \psi_n(i)$](img298.png)
![$ i$](img299.png)
![$ \psi_n$](img300.png)
are singular vectors of
![$ \hat{W}$](img302.png)
![$ \hat{W}^\ast \xi_n=\sigma_n \psi_n$](img303.png)
![$ \eta_n, n=1,\ldots,N$](img304.png)
the projector
![$ \hat{E}_{{\cal{V}}{\cal{W}^\bot}}$](img167.png)
Inversely, the representation (3) of
![$ \hat{E}_{{\cal{V}}{\cal{W}^\bot}}$](img167.png)
Proposition 2
The vectors
and
given in (11) and (12)
are biorthogonal to each other and span
and
, respectively.
The proof of this proposition can be found in [7]
Appendix A.
![$ \xi_n \in {\cal{W}}, n=1,\ldots,N$](img308.png)
![$ \eta_n \in {\cal{V}}, n=1,\ldots,N$](img309.png)
![$ {\cal{W}}$](img184.png)
![$ {\cal{V}}$](img39.png)
All the different numerical computations for an oblique projector discussed above can be realized with the routine ObliProj.m.