super symbol ($\s_l$ or $\x_l^a$) are i.i.d.

Similarly, we obtain $$\begin{split}
w_2=&\mathbb{E}_{\F,\z}[v_{\mu j}^a v_{\mu\eta}^{a}] = \frac{1}{N}{\sum_{l=1}^{N}}(s_{l,j}-x_{l,j}^a)(s_{l,\eta}-x_{l,\eta}^{a})\\
     &= \frac{1}{NJ}{\sum_{l=1}^{N}}(\s_l-\x_l^a)^T (\s_l^a-\x_l^b),\label{eq:sx_iid1}
\end{split}$$ where entries of $\s_l^a$ follow the same distribution as entries of $\s_l$ given $l$, and  is due to ([*i*]{}) entries of $\s_l$ being i.i.d., ([*ii*]{}) entries of $\x_l^{\{\cdot\}}$ being i.i.d. for fixed $l$, and ([*iii*]{}) the replica symmetry assumption [@Krzakala2012probabilistic; @krzakala2012statistical]. We also obtain $$\label{eq:v_j_eta_a}
\begin{split}
w_3=\mathbb{E}_{\F,\z}[v_{\mu j}^a v_{\mu j}^b]\!&=\!\frac{1}{NJ}\!{\sum_{l=1}^{N}}(\s_l\!-\!\x_l^a)^T \!(\s_l-\x_l^b)+\Delta.\\
w_4=\mathbb{E}_{\F,\z}[v_{\mu j}^a v_{\mu\eta}^{b}]&=\frac{1}{NJ}{\sum_{l=1}^{N}}(\s_l-\x_l^a)^T (\s_l^a-\x_l^b),
\end{split}$$

We now define the following auxiliary parameters $$\label{eq:auxParamsSet1}
\begin{split}
m_a=\frac{\displaystyle{\sum_{l=1}^{N}} (\x_l^a)^T\s_l}{NJ},&\quad Q_a=\frac{\displaystyle{\sum_{l=1}^{N}} (\x_l^a)^T\x_l^a}{NJ}, \\
q_{ab}=\frac{\displaystyle{\sum_{l=1}^{N}} (\x_l^a)^T\x_l^b}{NJ},&\quad
q_0=\frac{1}{NJ}{\sum_{l=1}^{N}}(\s_l^a)^T \s_l,
\end{split}$$ which allow us to express – as $$w_1=\rho-2m_a+Q_a+\Delta,$$ $$\label{eq:ws2}
w_2=q_0-(m_a+m_b)+q_{ab},$$ $$w_3 = \rho-(m_a+m_b)+q_{ab}+\Delta,$$ $$\label{eq:ws4}
w_4=q_0-(m_a+m_b)+q_{ab}.$$

Plugging the distribution of $\v_{\mu}$, approximated by $P(\v_{\mu})=[(2\pi)^n\det (\G_{\mu})]^{-\frac{1}{2}}\exp(-\frac{1}{2}\v_{\mu}^T\G_{\mu}^{-1}\v_{\mu})$, into , we obtain $$\label{eq:Xmu2}
\mathbb{X}_{\mu}=\l[\det(\mathbb{I}_n+\frac{1}{\Delta}\G_{\mu})\r]^{-1/2}.$$

[**MMV-2:**]{} For the matrix $\F$  in this setting, rows $jM+1,...,(j+1)M,\ 2\leq j \leq J$, will be the right-shift of rows $(j-1)M+1,...,jM$. We express $v_{\mu j}^a$  as $$\label{eq:v_mu_j_a_mmv2}
  v_{\mu j}^a={\sum_{l=1}^{N}}\F_{\mu l}{\bf T}_{j}(\s_l-\x_l^a)+z_{\mu+M(j-1)},\ \mu\in \{1,\cdots,M\},$$ where $\T_{j}$ is a $J\times J$ transform matrix with the $j$-th entry of the first row being one and all other entries in $\T_j$ being zeros. Using the same derivations as in MMV-1, it can be proved that the covariance matrix $\G_{\mu}=\mathbb{E}[\v_{\mu}\v_{\mu}^T]$ 