Journal article
On the Performance of Sparse Recovery Via ℓp-Minimization (0 < p ≤ 1)
IEEE transactions on information theory, Vol.57(11), pp.7255-7278
2011
DOI: 10.1109/TIT.2011.2159959
Abstract
It is known that a high-dimensional sparse vector x* in TV can be recovered from low-dimensional measurements y = Ax* where A m×n (m <; n) is the measurement matrix. In this paper, with A being a random Gaussian matrix, we investigate the recovering ability of ℓ p -minimization (0 ≤ p ≤ 1) as p varies, where ℓ p -minimization returns a vector with the least ℓ p quasi norm among all the vectors x satisfying Ax = y. Besides analyzing the performance of strong recovery where ℓ p -minimization is re quired to recover all the sparse vectors up to certain sparsity, we also for the first time analyze the performance of "weak" recovery of ℓ 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 ℓ 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 ℓ 1 -minimization. We also explicitly demonstrate that ℓ p -minimization (p <; 1) can re turn a denser solution than ℓ p -minimization. For any a G (0.1), we provide bounds of the sparsity ratio for strong recovery and weak recovery, respectively, below which ℓ p -minimization succeeds. Our bound of strong recovery improves on the existing bounds when a is large. In particular, regarding the recovery threshold, this paper argues that ℓ p -minimization has a higher threshold with smaller p for strong recovery; the threshold is the same for all p for sectional recovery; and ℓ p -minimization can outperform ℓ p -minimization for weak recovery. These are in contrast to traditional wisdom that ℓ p -minimization, though computationally more expensive, always has better sparse recovery ability than ℓ 0 -minimization since it is closer to ℓ 1 -minimization. Finally, we provide an intuitive explanation to our findings. Numerical examples are also used to un ambiguously confirm and illustrate the theoretical predictions.
Details
- Title: Subtitle
- On the Performance of Sparse Recovery Via ℓp-Minimization (0 < p ≤ 1)
- Creators
- MENG WANG - School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, United StatesWEIYU XU - School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, United StatesA O TANG - School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, United States
- Resource Type
- Journal article
- Publication Details
- IEEE transactions on information theory, Vol.57(11), pp.7255-7278
- Publisher
- Institute of Electrical and Electronics Engineers
- DOI
- 10.1109/TIT.2011.2159959
- ISSN
- 0018-9448
- eISSN
- 1557-9654
- Language
- English
- Date published
- 2011
- Academic Unit
- Electrical and Computer Engineering
- Record Identifier
- 9984083843202771
Metrics
22 Record Views