Highly Nonlinear Approximations for Sparse Signal Representation


Recursive updating/downdating of oblique projectors

Here we provide the equations for updating and downdating oblique projectors in order to account for the following situations:

Let us consider that the oblique projector $ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$ onto the subspace $ {\cal{V}}_k={\mbox{\rm {span}}}\{v_i\}_{i=1}^k$ along a given subspace $ {\cal{W}^\bot}$ is known. If the subspace $ {\cal{V}}_k$ is enlarged to $ {\cal{V}}_{k+1}$ by the inclusion of one element, i.e., $ {\cal{V}}_{k+1}= {\mbox{\rm {span}}}\{v_i\}_{i=1}^{k+1}$, we wish to construct $ \hat{E}_{{\cal{V}}_{k+1}{\cal{W}^\bot}}$ from the availability of $ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$. On the other hand, if the subspace $ {\cal{V}}_k={\mbox{\rm {span}}}\{v_i\}_{i=1}^k$ is reduced by the elimination of one element, say the $ j$-th one, we wish to construct the corresponding oblique projector $ \hat{E}_{{\cal{V}}_{k \setminus j}{\cal{W}^\bot}}$ from the knowledge of $ \hat{E}_{{\cal{V}}_k {\cal{W}^\bot}}$. The subspace $ {\cal{W}^\bot}$ is assumed to be fixed. Its orthogonal complement $ {\cal{W}}_k$ in $ {\cal{H}}_k= {\cal{V}}_k\oplus {\cal{W}^\bot}$ changes with the index $ k$ to satisfy $ {\cal{H}}_k= {\cal{W}}_k\oplus^\bot {\cal{W}^\bot}$.