Consider a complex vector $\mathbf{s} \in \mathbb{C}^{N \times 1}$ that is multiplied by a known unitary matrix $\mathbf{G} \in \mathbb{C}^{N \times N}$, yielding $\mathbf{y} = \mathbf{G}\mathbf{s}$. Suppose the entries of $\mathbf{s}$ at indices $\{1, 5, 9, 13, \ldots, N\}$ are known, while all other entries of $\mathbf{s}$ are unknown. Conversely, the vector $\mathbf{y}$ is fully known except for its entries at those same indices $\{1, 5, 9, 13, \ldots, N\}$. The goal is to recover the unknown portion of $\mathbf{y}$ (call it $\mathbf{y}_{o}$) using the known portion of $\mathbf{s}$ (denoted $\mathbf{s}_{k}$), given that $\mathbf{y}$ is corrupted by additive white Gaussian noise (AWGN). A direct approach uses linear algebra, where one might write:
$$ \mathbf{y}_{o} \;=\; \mathbf{G}_{k,k}^{-1}\,\mathbf{s}_{k} \;-\; \mathbf{G}_{k,k}^{-1}\,\mathbf{G}_{d,k}\,\mathbf{y}_{k},$$
with $\mathbf{G}_{k,k}$ being the square submatrix of $\mathbf{G}$ that multiplies the known entries $\mathbf{s}_{k}$, and $\mathbf{G}_{d,k}$ the submatrix coupling the rest of $\mathbf{y}$. Although this formula is correct in an ideal noise-free setting, any noise in $\mathbf{y}$ tends to get dramatically amplified by $\mathbf{G}_{k,k}^{-1}$, especially if $\mathbf{G}_{k,k}$ is ill-conditioned. (The submatrix $G_{k,k}$ is very small, that is almost zero).
How can we recover $\mathbf{y}_{o}$ without suffering severe noise amplification caused by directly inverting $\mathbf{G}_{k,k}$? Is there any other method to achieve that without amplifying the noise?
NP: $N$ is a $2^l$ integer number where $l$ is integer too.
