Let $\mathbf{X}$ be an $n\times n$ i.i.d. matrix with each entry being $1$ or $-1$ with probability $1/2$. Let $\mathcal{S}$ be a set, chosen at random with $\mathbb{E}|\mathcal{S}|=k$, such that each entry $(i,j)\in\{1,2,\ldots,n\}\times\{1,2,\ldots,n\}$ is included in $\mathcal{S}$ with probability $k/n^2$, independently. Finally, let $\mathbf{Y}_{\mathcal{S}}$ be the projection of $\mathbf{X}$ on $\mathcal{S}$, namely, for any $(i,j)\in\mathcal{S}$, $Y_{i,j}=X_{i,j}$, otherwise, $Y_{i,j}=0$.
I am wondering whether there are known upper bounds on the spectral (operator) norm of the matrix $\mathbf{Y}_\mathcal{S}$, and the $\ell$ moment spectral norm, $||\mathbf{Y}_\mathcal{S}||^\ell$.