Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compressed Sensing with Sparse Binary Matrices: Instance Optimal Error
  Guarantees in Near-Optimal Time   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.

11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin