  Many applications concern sparse signals, for example, detecting anomalies
from the differences between consecutive images taken by surveillance cameras.
This paper focuses on the problem of recovering a K-sparse signal x in N
dimensions. In the mainstream framework of compressed sensing (CS), the vector
x is recovered from M non-adaptive linear measurements y = xS, where S (of size
N x M) is typically a Gaussian (or Gaussian-like) design matrix, through some
optimization procedure such as linear programming (LP).
  In our proposed method, the design matrix S is generated from an
$\alpha$-stable distribution with $\alpha\approx 0$. Our decoding algorithm
mainly requires one linear scan of the coordinates, followed by a few
iterations on a small number of coordinates which are "undetermined" in the
previous iteration. Comparisons with two strong baselines, linear programming
(LP) and orthogonal matching pursuit (OMP), demonstrate that our algorithm can
be significantly faster in decoding speed and more accurate in recovery
quality, for the task of exact spare recovery. Our procedure is robust against
measurement noise. Even when there are no sufficient measurements, our
algorithm can still reliably recover a significant portion of the nonzero
coordinates.
  To provide the intuition for understanding our method, we also analyze the
procedure by assuming an idealistic setting. Interestingly, when K=2, the
"idealized" algorithm achieves exact recovery with merely 3 measurements,
regardless of N. For general K, the required sample size of the "idealized"
algorithm is about 5K.
