Highly Nonlinear Approximations for Sparse Signal Representation

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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:

i)

${\cal{V}}_{k \setminus j}={\mbox{\rm {span}}}\{v_i\}_{\genfrac{}{}{0pt}{}{i=1}{i \ne j}}^{k}={\mbox{\rm {span}}}\{v_i\}_{i=1}^{k}={\cal{V}}_k$ i.e., $v_j \in {\cal{V}}_{k \setminus j}.$

ii)

${\cal{V}}_{k \setminus j}={\mbox{\rm {span}}}\{v_i\}_{\genfrac{}{}{0pt}{}{i=1}{i \ne j}}^{k} \subset {\mbox{\rm {span}}}\{v_i\}_{i=1}^{k}={\cal{V}}_k$, 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:

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

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

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

This project is supported by EPSRC (EP/D062632/1)