of our algorithm for an SMV case and conjecture the optimality of our algorithm for MMV cases in Section \[sec:converse\]. For some example error metrics, we derive the corresponding optimal algorithms, together with the theoretical limits for these errors, in Section \[sec:example\]. Synthetic simulation results are discussed in Section \[sec:numeric\_synth\], followed by an application of our metric-optimal algorithm to a real-world problem in Section \[sec:app\]. We conclude in Section \[sec:conclusion\].

[**Notations:**]{} In this paper, bold capital letters represent matrices, bold lower case letters represent vectors, and normal font letters represent scalars. The $m$-th entry (scalar) of a vector $\z$ is denoted by $z_m$.

Problem Setting and Background {#sec:background}
==============================

Problem setting
---------------

[**Signal model**]{}: We consider an ensemble of $J$ signal vectors, $\x^{(j)}\in\mathbb{R}^N,\ j\in\{1,\ldots,J\}$, where $j$ is the index of the signal. Consider a [*super-symbol*]{} $\x_n=[x_n^{(1)},\ldots,x_n^{(J)}],\ n\in\{1,\ldots,N\}$; all super-symbols in this paper are row vectors. The super symbol $\x_n$ follows a $J$-dimensional distribution, $$\label{eq:jsm}
f(\x_n)=\rho \phi(\x_n)+(1-\rho)\delta(\x_n),$$ where $\rho\in (0,1)$ determines the percentage of non-zeros in the signal and is called the sparsity rate, $\phi(\x_n)$ is a $J$-dimensional pdf, and $\delta(\x_n)=\left\{
                \begin{array}{ll}
                 1, & \x_n = \textbf{0},\\
                 0, & \text{else}.
                \end{array}
            \right.\\
$

\[def:jointly\_sparse\] [*Ensembles of signals are called jointly sparse 