On the Performance of Sparse Recovery via

It is known that a high-dimensional sparse vector x∗ in Rn can be recovered from low-dimensional measurements y = Ax∗ where Am×n(m < n) is the measurement matrix. In this paper, with A being a random Gaussian matrix, we investigate the recovering ability of `p-minimization (0 ≤ p ≤ 1) as p varies, where `p-minimization returns a vector with the least `p quasi-norm among all the vectors x satisfying Ax = y. Besides analyzing the performance of strong recovery where `pminimization 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 `p-minimization (0 ≤ p < 1) where the aim is to recover all the sparse vectors on one support with a fixed sign pattern. When α(:= m n ) → 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 `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 `1-minimization. We also explicitly demonstrate that `pminimization (p < 1) can return a denser solution than `1minimization. For any α ∈ (0, 1), we provide bounds of the sparsity ratio for strong recovery and weak recovery respectively below which `p-minimization succeeds. Our bound of strong recovery improves on the existing bounds when α is large. In particular, regarding the recovery threshold, this paper argues that `p-minimization has a higher threshold with smaller p for strong recovery; the threshold is the same for all p for sectional recovery; and `1-minimization can outperform `p-minimization for weak recovery. These are in contrast to traditional wisdom that `p-minimization, though computationally more expensive, always has better sparse recovery ability than `1-minimization since it is closer to `0-minimization. Finally, we provide an intuitive explanation to our findings. Numerical examples are also used to unambiguously confirm and illustrate the theoretical predictions.