Let $\Phi$ be an $n \times N$ matrix, $\Phi_{I_{s-1}}$ be a subset of its column, $y=\Phi x_0$, $r_s=y-\Phi x_s$, $(x_s)_{I_{s-1}}=(\Phi_{I_{s-1}}^T\Phi_{I_{s-1}})^{-1}\Phi_{I_{s-1}}^{T}y$ where $(x_s)_{I_{s-1}}$ are the entries associated with $I_{s-1}$ and $(x_s)_{I^C_{s-1}}=0$. At the end of page 6 of Sparse Solution of Underdetermined Systems by Donoho he defines the $j$-th entry of the residual multi access inference as the following:
$z_s(j)=x_0(j)-\phi_j^Tr_s/||P_{I_{s-1}}\phi_j||_2^2, \quad j\notin I_{s-1}$
Question: From paper, what is $P_{I_{s-1}}$? Is it $P_{I_{s-1}}=(\Phi_{I_{s-1}}^T\Phi_{I_{s-1}})^{-1}\Phi_{I_{s-1}}^{T}$, correct me if I'm wrong. What is the intuition behind this definition? I mean how one would come up with this definition.