Highly Nonlinear Approximations for Sparse Signal Representation
Downdating the oblique projector
to
Let us suppose that by the elimination of
the element ![$ j$](img347.png)
![$ {\cal{V}}_k$](img343.png)
![$ {\cal{V}}_{k \setminus j}= {\mbox{\rm {span}}}\{v_i\}_{\genfrac{}{}{0pt}{}{i=1}{i \ne j}}^k$](img376.png)
![$ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}$](img348.png)
- i)
-
i.e.,
- ii)
-
, i.e.,
Case i)
Proposition 6
Let
be given by (17) and let us assume
that removing vector
from the spanning set of
leaves the identical subspace, i.e.,
.
Hence, if the remaining dual vectors are
modified as follows:
the corresponding oblique projector does not change, i.e.
.
![$ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$](img341.png)
![$ v_j$](img381.png)
![$ {\cal{V}}_k$](img343.png)
![$ v_j \in {\cal{V}}_{k \setminus j}$](img382.png)
the corresponding oblique projector does not change, i.e.
![$ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}=\hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$](img384.png)
Case ii)
Proposition 7
Let
be given by (17) and let us assume
that the vector
to be removed from the spanning
set of
is not in
.
In order to produce the oblique projector
the
appropriate modification of the dual vectors can be achieved
by means of the following equation
![$ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$](img341.png)
![$ v_j$](img381.png)
![$ {\cal{V}}_k$](img343.png)
![$ {\cal{V}}_{k \setminus j}$](img385.png)
![$ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}$](img348.png)
The proof these propositions are given in [10]. The code for updating the vectors are FrDelete.m.