Vector Problems'
---

[*Keywords*]{}:Active user detection, error metric, message passing, multi-measurement vector problem.

Introduction
============

Many systems in science and engineering can be approximated by a linear model, where a signal $\x \in \mathbb{R}^N$ is recorded via a measurement matrix $\A\in\mathbb{R}^{M \times N}$, and then contaminated by a measurement channel, $$\label{eq:SMV}
\w = \A \x,\ y_m = \mathcal{Z}(w_m), \forall m\in \{1,\ldots,M\},$$ where $y_m,m\in\{1,\ldots,M\}$, are the entries of the measurements $\y\in\mathbb{R}^M$, and the measurement channel $\mathcal{Z}(\cdot)$ is characterized by a probability density function (pdf), $f(y_m|w_m)$. The goal is to estimate $\x$ from the measurements $\y$ given knowledge of $\A$ and a model for the measurement channel $f(y_m|w_m),\forall m$. We call such a system the [*single measurement vector (SMV)*]{} problem.

In many applications, the signal acquisition systems are distributed, where $J$ measurement matrices measure $J$ different signals individually. The key difference between such a system and $J$ individual SMV’s, is that these $J$ signals are somewhat dependent. An example of a model containing such dependencies is the multi-measurement vector (MMV) problem [@chen2006trs; @cotter2005ssl; @Mishali08rembo; @Berg09jrmm; @LeeKimBreslerYe2011; @LeeBreslerJunge2012; @YeKimBresler2015]. The MMV problem considers the estimation of a set of dependent signals, and has applications such as magnetic resonance imaging with multiple coils [@JuYeKi07; 