Highly Nonlinear Approximations for Sparse Signal Representation


Downdating the oblique projector $ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$ to $ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}$

Let us suppose that by the elimination of the element $ j$ the subspace $ {\cal{V}}_k$ is reduced to $ {\cal{V}}_{k \setminus j}= {\mbox{\rm {span}}}\{v_i\}_{\genfrac{}{}{0pt}{}{i=1}{i \ne j}}^k$. In order to give the equations for adapting the corresponding dual vectors generating the oblique projector $ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}$ we need to consider two situations:
$ {\cal{V}}_{k \setminus j}={\mbox{\rm {span}}}\{v_i\}_{\genfrac{}{}{0pt}{}{i=1}{i \ne j}}^{k}={\mbox{\rm {span}}}\{v_i\}_{i=1}^{k}={\cal{V}}_k$ i.e., $ v_j \in {\cal{V}}_{k \setminus j}.$
$ {\cal{V}}_{k \setminus j}={\mbox{\rm {span}}}\{v_i\}_{\genfrac{}{}{0pt}{}{i=1}{i \ne j}}^{k} \subset {\mbox{\rm {span}}}\{v_i\}_{i=1}^{k}={\cal{V}}_k$, i.e., $ v_j \not\in {\cal{V}}_{k \setminus j}.$

Case i)

Proposition 6   Let $ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$ be given by (17) and let us assume that removing vector $ v_j$ from the spanning set of $ {\cal{V}}_k$ leaves the identical subspace, i.e., $ v_j \in {\cal{V}}_{k \setminus j}$. Hence, if the remaining dual vectors are modified as follows:

$\displaystyle {w_i}^{k \setminus j}= w_i^k+
 \frac{\langle u_j, w_i^k\rangle w_j^k}
 {1- \langle u_j, w_j^k\rangle },$ (21)

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}}$.

Case ii)

Proposition 7   Let $ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$ be given by (17) and let us assume that the vector $ v_j$ to be removed from the spanning set of $ {\cal{V}}_k$ is not in $ {\cal{V}}_{k \setminus j}$. In order to produce the oblique projector $ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}$ the appropriate modification of the dual vectors can be achieved by means of the following equation

$\displaystyle {w_i}^{k \setminus j}=w_i^k-\frac{w_j^k \langle w_j^k , w_i^k\rangle }{\Vert w_j^k\Vert^2}.$ (22)

The proof these propositions are given in [10]. The code for updating the vectors are FrDelete.m.