  This paper focuses studies the following low rank + sparse (LR+S) column-wise
compressive sensing problem. We aim to recover an $n \times q$ matrix, $\X^* =[
\x_1^*, \x_2^*, \cdots , \x_q^*]$ from $m$ independent linear projections of
each of its $q$ columns, given by $\y_k :=\A_k\x_k^*$, $k \in [q]$. Here,
$\y_k$ is an $m$-length vector with $m < n$. We assume that the matrix $\X^*$
can be decomposed as $\X^*=\L^*+\S^*$, where $\L^*$ is a low rank matrix of
rank $r << \min(n,q)$ and $\S^*$ is a sparse matrix. Each column of $\S$
contains $\rho$ non-zero entries. The matrices $\A_k$ are known and mutually
independent for different $k$. To address this recovery problem, we propose a
novel fast GD-based solution called AltGDmin-LR+S, which is memory and
communication efficient. We numerically evaluate its performance by conducting
a detailed simulation-based study.
