English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is known that a high-dimensional sparse vector x* in R^n can be recoveredfrom low-dimensional measurements y= A^{m*n} x* (m<n) . In this paper, weinvestigate 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 ``norm'' among allthe vectors x satisfying Ax=y. Besides analyzing the performance of strongrecovery where l_p-minimization needs to recover all the sparse vectors up tocertain 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 allthe sparse vectors on one support with fixed sign pattern. When m/n goes to 1,we provide sharp thresholds of the sparsity ratio that differentiates thesuccess and failure via l_p-minimization. For strong recovery, the thresholdstrictly 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 thresholdis 1 for l_1-minimization. We also explicitly demonstrate that l_p-minimization(p<1) can return a denser solution than l_1-minimization. For any m/n<1, weprovide bounds of sparsity ratio for strong recovery and weak recoveryrespectively below which l_p-minimization succeeds with overwhelmingprobability. Our bound of strong recovery improves on the existing bounds whenm/n is large. Regarding the recovery threshold, l_p-minimization has a higherthreshold with smaller p for strong recovery; the threshold is the same for allp for sectional recovery; and l_1-minimization can outperform l_p-minimizationfor weak recovery. These are in contrast to traditional wisdom thatl_p-minimization has better sparse recovery ability than l_1-minimization sinceit is closer to l_0-minimization. We provide an intuitive explanation to ourfindings and use numerical examples to illustrate the theoretical predictions.

On the Performance of Sparse Recovery via L_p-minimization (0<=p <=1)