$X \leq CY$ for all sufficiently large $N$ and for some constant $C$. Notations such as $X \ll_k Y$ and $X=O_k(Y)$ mean that the hidden constant $C$ depends on another constant $k$. We always allow the implicit constants in our asymptotic notation to depend on the integer $m$ from Theorem \[thm:main\]; we will not denote this dependence with a subscript. $X=o(Y)$ or $Y=\omega(X)$ means that $X/Y \rightarrow 0$ as $N \rightarrow \infty$.

$\|{\mathbf}{M}\|$ is the spectral norm of the matrix ${\mathbf}{M}$. $\|{\mathbf}{M}\|_2$ denotes the Hilbert-Schmidt norm of ${\mathbf}{M}$ (defined in ). We let ${\mathbf}{I}_N$ denote the $N \times N$ identity matrix. Often we will just write ${\mathbf}{I}$ for the identity matrix when the size can be deduced from the context.

We write a.s., a.a., and a.e. for almost surely, Lebesgue almost all, and Lebesgue almost everywhere respectively. We use $\sqrt{-1}$ to denote the imaginary unit and reserve $i$ as an index. We let ${\ensuremath{\mathbf{1}_{{E}}}}$ denote the indicator function of the event $E$.

We let $C$ and $K$ denote constants that are non-random and may take on different values from one appearance to the next. The notation $K_p$ means that the constant $K$ depends on another parameter $p$. We always allow 