Highly Nonlinear Approximations for Sparse Signal Representation
Updating the oblique projector
to
We assume that
![$ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$](img341.png)
In order to inductively construct the duals
![$ {w}^{k+1}_i, i=1,\ldots,k+1$](img354.png)
- i)
-
, i.e.,
- ii)
-
, i.e.
Case i)
Proposition 4
Let
and vectors
in (17) be given. For an arbitrary vector
the dual vectors
computed as
for
and
produce the identical oblique projector as the dual vectors
.
![$ v_{k+1} \in {\cal{V}}_k$](img359.png)
![$ w_i^k$](img360.png)
![$ y_{k+1}\in{\cal{H}}$](img361.png)
![$ w_i^{k+1}$](img362.png)
for
![$ i=1,\ldots,k$](img364.png)
![$ w^{k+1}_{k+1}=y_{k+1}$](img365.png)
![$ w_i^k, i=1,\ldots,k$](img366.png)
Case ii)
Proposition 5
Let vector
and vectors
in (17) be given. Thus
the dual vectors
computed as
where
with
,
provide us with the oblique projector
.
The proof these propositions are given in [10].
The codes for updating the dual vectors are
FrInsert.m
and FrInsertBlock.m.
![$ v_{k+1} \notin {\cal{V}}_k$](img367.png)
![$ w_i^k, i=1,\ldots,k$](img366.png)
![$ w_i^{k+1}$](img362.png)
where
![$ w_{k+1}^{k+1}= \frac{\gamma_{k+1}}{\Vert\gamma_{k+1}\Vert^2}$](img369.png)
![$ \gamma_{k+1}= u_{k+1} - \hat{P}_{{\cal{W}}_k} u_{k+1}$](img370.png)
![$ \hat{E}_{{\cal{V}}_{k+1}{\cal{W}^\bot}}$](img346.png)
Property 2
If vectors
are linearly independent
they are also biorthogonal to the dual
vectors arising inductively from
the recursive equation (19).
The proof of this property is in [10].
![$ \{v_i\}_{i=1}^k$](img371.png)