  A compressed sensing method consists of a rectangular measurement matrix, $M
\in \mathbbm{R}^{m \times N}$ with $m \ll N$, together with an associated
recovery algorithm, $\mathcal{A}: \mathbbm{R}^m \rightarrow \mathbbm{R}^N$.
Compressed sensing methods aim to construct a high quality approximation to any
given input vector ${\bf x} \in \mathbbm{R}^N$ using only $M {\bf x} \in
\mathbbm{R}^m$ as input. In particular, we focus herein on instance optimal
nonlinear approximation error bounds for $M$ and $\mathcal{A}$ of the form $ \|
{\bf x} - \mathcal{A} (M {\bf x}) \|_p \leq \| {\bf x} - {\bf x}^{\rm opt}_k
\|_p + C k^{1/p - 1/q} \| {\bf x} - {\bf x}^{\rm opt}_k \|_q$ for ${\bf x} \in
\mathbbm{R}^N$, where ${\bf x}^{\rm opt}_k$ is the best possible $k$-term
approximation to ${\bf x}$.
  In this paper we develop a compressed sensing method whose associated
recovery algorithm, $\mathcal{A}$, runs in $O((k \log k) \log N)$-time,
matching a lower bound up to a $O(\log k)$ factor. This runtime is obtained by
using a new class of sparse binary compressed sensing matrices of near optimal
size in combination with sublinear-time recovery techniques motivated by
sketching algorithms for high-volume data streams. The new class of matrices is
constructed by randomly subsampling rows from well-chosen incoherent matrix
constructions which already have a sub-linear number of rows. As a consequence,
fewer random bits than previously required are needed in order to select the
rows utilized by the fast reconstruction algorithms considered herein.
