On the Performance of Sparse Recovery Via $\ell_p$-Minimization $(0 \leq p \leq 1)$

It is known that a high-dimensional sparse vector
${\bf x}^*$
in
${\cal R}^n$
can be recovered from low-dimensional measurements
${\bf y}=A{\bf x}^*$
where
$A^{m \times n}(m < n)$
is the measurement matrix. In this paper, with
$A$
being a random Gaussian matrix, we investigate the recovering ability of
$\ell_p$
-minimization
$(0\leq p \leq 1)$
as
$p$
varies, where
$\ell_p$
-minimization returns a vector with the least
$\ell_p$
quasi-norm among all the vectors
${\bf x}$
satisfying
$A{\bf x}={\bf y}$
. Besides analyzing the performance of strong recovery where
$\ell_p$
-minimization is required to recover all the sparse vectors up to certain sparsity, we also for the first time analyze the performance of “weak” recovery of
$\ell_p$
-minimization
$(0\leq p < 1)$
where the aim is to recover all the sparse vectors on one support with a fixed sign pattern. When
$\alpha (:={m \over n}) \rightarrow 1$
, we provide sharp thresholds of the sparsity ratio (i.e., percentage of nonzero entries of a vector) that differentiates the success and failure via
$\ell_p$
-minimization. For strong recovery, the threshold strictly decreases from
$0.5$
to
$0.239$
as
$p$
increases from
$0$
to
$1$
. Surprisingly, for weak recovery, the threshold is
$2/3$
for all
$p$
in
$[0,1)$
, while the threshold is
$1$
for
$\ell_1$
-minimization. We also explicitly demonstrate that
$\ell_p$
-minimization
$(p < 1)$
can return a denser solution than
$\ell_1$
-minimization. For any
$\alpha \in (0,1)$
, we provide bounds of the sparsity ratio for strong recovery and weak recovery, respectively, below which
$\ell_p$
-minimization succeeds. Our bound of strong recovery improves on the existing bounds when
$\alpha$
is large. In particular, regarding the recovery threshold, this paper argues that
$\ell_p$
-minimization has a higher threshold with smaller
$p$
for strong recovery; the threshold is the same for all
$p$
for sectional recovery; and
$\ell_1$
-minimization can outperform
$\ell_p$
-minimization for weak recovery. These are in contrast to traditional wisdom that
$\ell_p$
-minimization, though computationally more expensive, always has better sparse recovery ability than
$\ell_1$
-minimization since it is closer to
$\ell_0$
-minimization. Finally, we provide an intuitive explanation to our findings. Numerical examples are also used to unambiguously confirm and illustrate the theoretical predictions.