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Given the matrix \(P\) or equivalently the basis \(\{\phi_i\}\), compressed sensing allows to reconstruct the full matrix \(A\) sampling \texit{some} of the entries of \(B\). The reconstruction is done by solving the basis pursuit problem (BP),  \begin{align}  \label{eq:bpdn}  \min_{A} ||A||_1 \quad \textrm{subject to} \quad \(PAP^T\)_{ij} (PAP^T)_{ij}  = B_{ij}\quad \forall  i,j \in R\ ,  \end{align}  where the 1-norm is considered as a \emph{vector} norm  (\(||A||_1 = \sum_{i,j} \left|A_{ij}\right|\)).